Power Series (ultimate study guide)

okay let's do some math for fun  today we'll be doing 26.2 power   series representation of functions and of course  we also find out their interval of convergence   and yes we will do everything in one take and of  course I have the file right here for you guys   with all the equations and I also have the answer  key I will be checking the answer key later on   along the way just to make sure I didn't make any  typos and for the people were taking file - well   don't forget this is what you need right how you  test out if a series converges or not and on the   back here you have the best friend and also all  the other friends that you should know okay so all   the things right here will be in the description  for you guys and this video is dedicated to all   the students what you can Cal - and also for  the people who need to know about power series   okay so here we go first of all you need to know  your best read read it really well for example   the first question we have x over 1 - works I will  tell you guys the function right here and then the   center and we'll be using Sigma notation to write  down the power series alright anyway for the best   friend for this one right let me just write this  down yeah for the best friend well 1 over 1 minus   X this is our best friend named leader geometry  series but we like the best friend much better   this is equal to the series as n goes from 0  to infinity X to the nth power and this only   works if the absolute value of x is less than 1 so  include this inequality because it will tell you   about the radius of convergence based on this we  know R is equal to 1 and because the center says 0   and R is equal to 1 that means we can move to the  left and also the right one unit each direction   so we know the interval convergence will be going  from negative 1 to 1 and you also to remember that   for the best friend at the in tat the endpoints  of the I where we do not include the endpoints   because it won't come purge and let's take a look  of how to do this one here we go I will recommend   you guys to read this down right here it's over  1 minus works and now we see that in water for   yourself best friend we need to have a point on  the top but this is next not so good that's okay   because we can just put the eggs in the front like  so then we can just look at this s times one over   and notice that for the best friend we must have  one - and this is one - so that's perfect we have   the phone that we want and this right here is just  a 4x and in fact that would be our new input now   let me show you this X stays in a front but 1 over  1 minus 2x we can just enter leaf works right here   because you see the best friends 1 over 1 - while  reading who you have just put here we can do the   same right here and we can just do some algebra  on this right so let's see this is going to give   us the series as n goes from 0 to infinity and by  the way I may say this as a series who I may also   say this as a sum they are interchangeable right  anyway you put 14 here open the parentheses and   raised to the nth power and don't forget to put  it works in here so we have the absolute value   of 4x that has to be less than 1 like this then we  are just going to fix this pretty quick that's all   so we have the X in the front and this is just  a series as n goes from 0 to infinity 4 to the   N power times X to the nth power yeah and in  the end this is X and don't forget that this   right here is infinite polynomial and we can just  distribute the X right here so in the end we get   the series as n goes from 0 to infinity and this  is 4 to the N and this is X - D this is the first   power and this is nth power so you just add the  exponents and that's the form now we have and I   also recommend you guys always try to put the  X to some power at the end like this I have a   coefficient right here okay then look at this  we have to simplify this a little bit absolute   for your voice is the same as 4 times absolute  5 X is less than 1 like this divide both sides   by 4 we get absolute value of x is less than one  over four with this being done we know that the   radius of convergence is just one over four and  then the in terms of convergence it's going to be   well from zero we move to a lab and also move to  right 1 over 4 units each time so we have 1 over   4 negative and 1 over 4 past hip and now pay close  attention to this notice how we just did algebra   with our best friend and that was this right in  that case we also do not include the endpoints   of the interval of convergence so this and that  maintain parentheses right main tend to be open   like this only later on when we are doing like  integrations or maybe differentiations we really   have to be careful with the endpoints of the  eye by just doing algebra it stays the same time   right so this is it it wasn't so bad isn't it told  you best brand will help us out tremendously yep   that's the first one and let's see I'm just going  to save some space so I'll just register ok so   this is the R is equal to 1 is equal to negative  1 1 number 2 here we go we have X to the fourth   power over 9 plus X square and we want the center  to be at a equal to 0 later we'll see what if we   have a different center it may be more bizarre  but for now just a 0 ok here we go of course let's   write down our best friend first this is 1 over  1 minus X and this is the sum as n goes from 0 to   infinity X to the nth power absolute 5 X is less  than 1 and just keep this in mind I'm not going   to write this down anymore for the best friend or  is 1 and this is the I now this is what we have we   have let me just erase this up more space right ok  this is what we have we have X to the fourth power   over 9 plus x squared we will be doing the similar  scene first of all this is X to the fourth power   we don't like it it's ok let me just put it on the  side X to the first power so we can do Katie like   X for fourth power times one now on the bottom we  want to have a one but this is a nine not so good   but don't worry small fixed right because look  at this and that we can just factor or not so   I will ready Dallas this overnight and then put  on parenthesis like this originally we had an i   we factor DL so we have a one well this right  here it's a plus but in water for use two out   you use of best friend we have to have a - it's  okay because a plus is the same as minus negative   still the same as that then X square we factor the  other nine so it's X square don't forget to tip a   tip I night like this alright very good let's put  little parenthesis like so here we will have X to   the fourth power divided by nine same business  throwing this right here so we have the series   as n goes from zero to infinity open a parenthesis  with negative x squared over nine and raised to   the nth power and don't forget to throwing this  into the absolute value negative x squared over   nine make this equal make this less than 1 and now  we'll just have to fix this real quick this right   here is X to the fourth power over nine and this  is the sum as n goes from zero of nasty as n goes   from zero to infinity now let's do the following  we have negative 1 to the nth power that's write   that down first so here we have negative 1 to  the nth power on the bottom we have nine to the   nth powers or IDs 9 to the nth power and then this  is X square to the nth power so radius X to the 2   1 yeah and then just do the outro for that focus  on this we will distribute the X to a 4 and also   the 9 so it looks like we have the series as  n goes from 0 to infinity this is negative 1   to the nth power over this and that is 9 n plus 1  power this and that is x2 to add plus horsepower   like this very very nice okay now just work this  out real quick absolute value the negative doesn't   matter inside and when you have absolute value  of x squared it's a same as absolute value of   x and then squirrel on the other side right so  this is the same si up to 5x squared overnight   it's less than 1 then of course we can multiply  the now both sides this is up to 5x squared is   less than 9 and then take the square root on both  sides don't worry about the plus minus and this   is not yet because the absolute value of x this  part right here it's always no non-negative so   this is good and in the end of course we can just  isolate the apps to 5x which is going to be less   than 3 what's this being done we know that R is  equal to 3 and then because the center is at 0   that means the I the interval of convergence  will go from negative 3 to 3 and again do   not include endpoints because we just did algebra  without bedspread so this right here would be the   answer for number 2 that's it and we just have  twenty four point two to go I'm sorry it's now   100 questions per you know I think this right here  should be efficient effective for you guys so yeah and I do plan to make 100 calculus two questions  for you guys by the end of this year and of course   by before the final exam of course yeah so  you just can expect that I don't know when   but hopefully sometimes during the next week or so  I should just focus instead of talking I trying to   write okay number three we have 1 plus 2x over  man one past works 1 minus X at a equal to 0 hmm   okay let's see how we can do this of course that's  go ahead and put our best friend I'll just write   down a few more times for you guys 1 over 1 minus  X this right here it's the sum as n goes from 0   to infinity X to the nth power and after the four  bags has to be less than 1 now this right here's   what we have yeah so let me just read it out 1  plus 2 x over 1 minus X for the best friend we   only have 1 turn on the top name is just 1 but we  have two terms it's ok just separate them right   so this right here we can look at this as 1 over 1  minus X for the first fraction and for the second   fraction we put here 2x over that and we add 2x  over 1 minus X like this well this is our best   breast chopped right this is not quite but almost  this radius small fix because you know we can just   write as 1 over 1 minus X here M plus quite yes  2x times 1 over 1 minus X like this very nice then   we'll see this right here is directly our best  friend so this is the series as n goes from 0   to infinity of X to the nth power and then this is  also a hole best friend straight right here right   so this is just going to be the 2x right here and  we have the series as n goes from 0 to infinity   and this is just experienced power nothing we're  and because we went from here to here by using   the best friend so be sure now we just enter the  absolute power of X is less than 1 and the input   right here is just X so is the same one right here  okay now multiply this and that so it looks like   we have the series as n goes from 0 to infinity  X to the nth power plus well already down is the   sum and goes from 0 to infinity this is X to the  first power so we will have X to the n plus 1 and   this is of course just a 2 I shall have told you  like this so two times just a tool like this hmm   it seems okay but can we combine the terms things  the answer to that yes we can this is how so let   me show you first thing first in order for you to  combine anything deep hole right here extra match   this is X 2 and this is X to the n plus 1 this is  not good so this is what you do with this series I   will maintain the same one right here the series  as n goes from 0 to infinity X to the nth power   what we are going to do is we are just going to  subtract 1 from the end here so we have 2 to the   X I mean 2 times X to the nth power only like this  but we subtracted one more how can we balance this   out well just go here in the Sigma notation we  add 1 we add 1 that's how we shift it and keep   things balanced so here we will have the sum as n  goes from 1 to infinity plus once during affinity   again minus 1 to the N but I want add 1 so that's  what we have and now you see that the index right   this is going from 0 to infinity this is going  from 1 to infinity so they don't match anymore   hmm it's ok easy fix remember what this is this  means you put in 0 and you add you put in 1 and   you're putting 2 and so on right and this equals  0 1 2 3 and so on so this is what we'll do I will   just show you if we pelt if we put an equal to  0 well that's just the first 10 so we just get   1 this is when n is equal to 0 okay because X plus  2 roots power we can want and of course next it's   just of what 9 is equal to 1 and you get X and  then to the first power and then the next one   is just put 2 in it you have x squared and so on  so on and so on yeah and this is what you do you   leave it right here now you leave this right here  this one stays but this this right here you put it   in the Sigma notation so it's going to look like  the sum as n goes from 1 to infinity X to the nth   power like this and again this right here is for  the first part I should have done that in purple   but like let me just indicate that this right here  it's the first part okay then for the second part   just go ahead and maintain that we have the sum  or series n goes from 1 to infinity and we have   the 2 X to the nth power like this and now when  the index 1 goes 1 to infinity 1 to infinity are   same this is X to the N X 4 and of course we can  just put things together therefore we have the 1   all the way in the front and we have the sum and  this is the and goes from 1 to infinity 1 plus 2   is 3 and then X to the nth power like this yeah so  this right here will be the form for that when you   combine as much as possible okay and of course  we didn't do anything weird this is all we have   that means R is equal to 1 and I is negative 1 to  1 do not include the endpoints and that's it this   right here it's actually a pretty good technique  if you are doing this for differential equations   again you fix the power here and then you are  going to fix the index and by to fixing the index   you just write a few terms you can just join zeros  right here and then this is just going to be from   1 to infinity of the same thing yeah like this ok  that was question number 3 now question number 26   already know still a few more to go I don't know  how long this will take me not because I want to   give you guys some details explanation because I  know power series are weird most of the times oh   yeah next one number 4 here here we have 1 over x  squared minus 5x minus 6 and the center a is zero   man this is not a best friend doesn't look like  the best friend at all and the worst thing is   that yes results best friend your sauce only has  two right but we do notice that this right here x   squared minus 5x minus six we can factor it isn't  it this right here we can factor this is X plus 1   times X minus 6 that's good and if we do that  what can we do next partial fractions yeah so   here we go partial fraction for this guy okay for  the first one I'll just write years X plus 1 some   number of that and a Plus this one is X minus 6  and all that now here we go to figure out this   number we go back to the original and comprar the  same denominator and ask yourself how can we make   X plus 1 equal to 0 where X has to be negative 1  and then we put negative 1 in here so we have 1   1 over negative 1 minus 6 namely we get negative  1 over 7 all right and then we do the same thing   here but cover this up we need to make X equal to  6 put in here so we have 1 over 6 plus 1 namely we   get 1 over 7 like this ok and we know our best  friend already and let's see hmm okay which is   the tincture our best friend right here so here  I will just write yes negative 1 over 7 all the   way in the front and multiplied by this is one  Tapatio alpha of course we can switch the water   no big deal so we have one but this is a plus no  big deal because plus is the same as minus minus   so we have minus minus X so that would be for  the first part in one ready for using our best   friend next we add this is 1 over 7 let's put it  on the side like this and then we have the 1 over   man we have a X minus 6 we need to have a number  goes first right because this is one minus X so   we will have to switch the water of subtraction  in the meantime need to make this one not a six   it's okay we will just use a purple pen to take  your business for us today is yeah purple paint   because unfortunately with enough blue paint in my  department so yeah one over seven okay here we go   I'm going to just have a one over that looks more  better anyway we factor out the negative 6 on the   bottom like this and we see that okay to create  this done carefully originally we had X minus 6   but we fight other than negative 6 so we have one  okay and when we factor out a negative six let me   just put this down when we write our negative six  when we factor out negative is six minus X right   but then yeah but then I also want to factor  out the negative sort of a total six so this   is negative six and this is one minus x over six  like this okay so this is going to be one minus   x over six like that so again I switch the water  by factoring on negative and then factor the 6 and   then 1 minus x over 6 and that's what we have and  of course you can always double-check triple-check   just by multiplying this times that is negative  6 this time that is past d-bags so we did it okay   now you saw path spread this right here we get  negative 1 over 7 or during the front and this   right here we have the sum and goes from 0 to  infinity open a parenthesis you enter this into   the parenthesis that it just opened yeah and then  right here okay multiply this out we get minus or   put this time purple minus 1 times 1 is 1 and then  42 and here we join this into that parenthesis   and goes from 0 to infinity and here we have a  parenthesis x over 6 it's just pass through okay   to the nth power like this whoo pretty good now  just do the rest yeah so here we have negative   one over seven this is the sum and goes from zero  to infinity write this as negative 1 to the nth   power times X to the nth power like this for this  one man I will still use the purple pen so this   is a minus 1 over 42 and then here we have the  sum and goes from zero to infinity and this is   1 over 6 to the nth power you should write down  the coefficients first and then write down X to   the nth power on the side like that yeah and now  we were just kind of drawing everything because   this right here of course we can just change this  put it inside multiply and then likewise we also   do the same right here now we do two things in  one step here we go both of them they have the   samp going from 0 to infinity so we can just write  this down as one series in the one Sigma notation   first of all we have this which is negative 1 over  7 times da so I shall have this right here on the   top negative 1 to N power ok and then because  both of them will have the extras power so I'm   just going to put that at the end next we have  this times that so perhaps I'll just read this   as minus in purple and one on top of course that's  just put it on black and over this is 42 so we'll   have the 42 right here times X times 6 to the nth  power and then as I said this is pretty much the   coefficient and we multiply by X to the nth power  like this and I know I haven't put on absolute   value yet but I just want to make sure that my  answer key I give you guys some legitimate stuff yeah kind of us you were saying yeah kind of  polite it really depends a Hodgins simplified   this guy okay so you just leave it if you would  like you don't like it man I don't know even if   you leave like this is totally okay seriously okay  by the way right here you need two conditions the   first condition is that along here to here you  enter the negative X into the best friend right   so the first thing is that you look at absolute  value of negative x here is less than one and   you also have to make sure because we entered  it x over 6 into the best breath so we have   absolute value of x over 6 to be less than 1 and  of course you are just going to look at this and   I just kind of solve to see taking equality this  right here is straightforward up to 5 X is less   than 1 and this right here well multiply the  six ample set up to 12 X is less than 6 well   this is smaller right this is himself that so  you just care about this one because of the end   situation this and that so this is yet so that  means all will be just 1 because yeah this is   from one negative 1 to 1 this is yeah inside  hmm so I'm gonna leave it right here but I do   have perhaps I just an algebra person of the  answer on the air so if you guys were like you   guys can try your best to enter that I think  I'm done for this so just how you are again 1   okay and I will just go from negative 1 1 making  this encoder so oops so this right here is it right yeah yeah okay so that's question number  four and now let's see if I can screw see me   another one question number five okay I think I  can do this one okay one over one minus x squared   number five this is 1 over 1 minus X and then  square Center the add a equals zero man if the   question was one over one minus x squared this  will happen so much easier because all we have   to do is to enter the X square into our past right  into the summation right here right that's all but   unfortunately now this is what we have 1 over 1  minus X in parentheses and then we square that   of course this is a 7 saying 1 over parenthesis  1 minus X to the 2nd power if you want to expand   this maybe but maybe not don't do it so now this  is what we are going to do okay so let me actually   just write this down let me just actually try this  down one over parenthesis 1 minus x squared that's   actually hard loading the question actually  at a equal to 0 anyway hmm let's see what's   a connection freaking out best friend ah it seems  is just the best friend square but if you put that   down you have the sum as X this is the best friend  square don't do that it's much harder life doesn't   have to be that hard sometimes this is how we're  going to do it so here we go I'm Johnson write   this down again again I'll just write down the  best brand' 1 over 1 minus X which is we know let   me just change it doesn't matter in this case so  which we know this is the sum as n goes from 0 to   infinity we have X to the nth power and absolute  value of x is less than 1 okay well the bottom is   being squared the truth is if we differentiate the  best fret we can actually get the square on the   bottom let me show you so on the left hand side  how to differentiate the best friend with respect   to X likewise on the right hand side will also  differentiate this right okay to differentiate   this of course we look at this as negative power  so we have 1 minus X raised to the negative 1 it's   much easier this way so let's focus on defense  that first well to differentiate this we can of   course bring the power to the front and then minus  1 so we get negative parenthesis with 1 minus X in   here and then that's the negative 2 power but  don't forget the chandu x leader of the inside   the rupee of 1 minus X is negative 1 so multiply  by negative 1 so in fact negative negative becomes   a positive so we get 1 over parentheses 1 minus  x squared like this very nice very nice right so   that's the idea I want you to differentiate this  you get that and let me just erase this so I have   more space when we differentiate that we get 1  over premises 1 minus X square and congratulations   on the right hand side to differentiate a power  series you don't need to use like anything tricky   okay unfortunately we can use the chain nope I  didn't and sometimes this it's better that way   differentiating power series go ahead and bring  the power to the front and a minus 1 and got done   that's all so I will write this down we will have  the sum as n goes faster number to infinity and   let me just write down an X to the N minus 1 power  like this now here's the deal if I put down when   n is equal to 0 right because if I maintain it  well if we push the row into the end right here   and right here first we will just get 0 times that  it's just a rope so it's nothing if you put the   world right here ok it's not wrong but change it  we go with 1 because the first term matters when   n star with one that's the only term that's the  first term number actually matters and then of   course 2 3 4 5 so on to infinity like this and  right here I will also tell you we know another   thing you gotta know when we differentiate the  power series the radius of convergence stay the   same the radius of convergence stay the same right  so earlier the best friend always won now R was   still be won but for the interval of convergence  you will just have to double check well how do we   check it at putting negative 1 in here so it looks  like let me just show you check when X is equal to   negative 1 you put it right here so we are looking  at some as n goes from 1 to infinity and here we   have n times negative 1 let me just change this  in red X is equal to negative 1 to the negative 1   here minus 1 yeah like this now the question is if  n goes to infinity does this converge no because   if you look at the am part right here the limit as  n goes to infinity of n times negative 1 to the N   minus 1 power this right here what does not exist  so in fact this right here diverges by test for   diverges so right here we still do not include the  endpoint similarly we are going to check when X is   equal to 1 check when n is equal to 1 you put one  right here so we're looking at the sum and goes   from 1 to infinity and it's right here and then  1 to the N minus 1 power which it doesn't matter   and again this right here the limit as n goes to  infinity of just this this right here against na   equal to 0 so diverges by test for diverges you  also don't include the other endpoint right so   that's number 5 yeah okay I think the markers are  not too weird today because it's kind of hard for   me to clean the port but it's okay I'm doing this  for you guys that was number fire now we are going   to take this into a different style number six  here we go natural log of 1 plus X man natural   log and this time at the center again zero don't  worry Dayton will be doing the different Center   first thing first is there any connection between  a best friend and a natural log it seems like it   because if we integrate 1 over a linear function  we can get natural log right so this is what we   are going to do let's look at the best friend  again we know that this is 1 over 1 let me just   write on everything black so we have worked with  this case I need read earlier I didn't either read   but yes this to this is the best friend n goes  from 0 to infinity parenthesis I mean X to the   nth power like this absolute value of x is less  than 1 right now I need the best friends cousin   because the input here is 1 plus X I'm not going  to integrate this right here since okay though   I will just write this top 1 over 1 plus X this  is the same as 1 over you know the deal 1 minus   parenthesis negative x so I can put a negative x  here this is the sum and goes from 0 to infinity   parenthesis negative x to the nth power yep and  then of course don't forget to put a negative x   in the absolute value to be less than 1 now we  are going to focus on this the defense sign is   1 over 1 plus X the right hand side this is the  sum and goes from 0 to infinity and again just   cannot separate the coefficient and also X to the  power this is negative 1 to the nth power and we   have X to the nth power like this now purple pens  time we are going to integrate both sides go ahead   and do it right here and right here with respect  to X namely in the X world on the left hand side   when we integrate 1 over linear function we do end  up with the natural log won't want very nice well   natural log and you don't need absolute value  because we are on this interval absolute value   of past the X will be less than 1 so this 1 plus  X will always be positive in that interval so we   just need regular parentheses 1 plus X like so  don't worry about the plus C just worry about a   plus you can add right hand side but you know we  are going to get a summation like this right I'm   not going to put on a plasti at the end because  sometimes this might look like it's belonging   inside of the summation well the way to fix Dyess  you put down the plus verse okay now integrate   this guy we still have the summation this time we  don't use any terms we will still have and goes   from 0 to infinity now to integrate power series  is just as easy because all we have to do is to   get a power right here which was at 1 and that's  a new power and they come here divided by the new   power that's it very nice so here we have the  coefficient namely negative 1 to the nth power   over n plus 1 and then we have X to the n plus  1 power yeah love integrate things power series   who does it right ok so that's pretty much Jenna  will report us they will worry about this later   however though we still have to find out what  the series to do so because the left hand side   is a known function so we can do the following I'm  just going to pick a easy enough x-value first use   0 of course because that way the left hand side  is going to be Ln of 1 plus 0 and that of course   just 0 now the right hand side this is C I will  enter the 0 here so we plus sum as n goes from   zero to infinity write this down negative one  to the N power over n plus one and then put a   symbol right here so we have zero to the n plus  1 power the left hand side is zero right there   already and this is C plus well well when n is  zero that's the first term put it here you have   to go to the first power so that means that term  is zero and then when n is 1 again you get 0 and   so on so on so on you get infinitely many solid  exact zeros add them up you get zero so this is   just going to give you zero in another word C is  just nice to equal to zero in another word this   D is just equal to zero oops C is equal to C were  obvious no no it looks like C is equal to e there   we go ok so C is equal to zero finally I can come  here and tell you the apps not the absolute value   Ln of 1 plus X this right here is nicely equal  to just this part the sum as n goes from 0 to   infinity and we have negative 1 to the nth power  over n plus 1 times X to the M plus ones power   like this very nice now our well when we integrate  what differentiate our series the R will stay the   same earlier we integrate the best friends cousin  which the R is still you know less absolute value   is 1 less than 1 so are still equal to 1 R is  1 Center was a 0 so we go from negative 1 to 1   for the I let me just write some better I will be  going from negative 1 to 1 but I don't know about   the endpoints it's not parentheses anymore you'll  see again we will have to do what with the earlier   collection for the check let me check negative  1 right so I will just tell you we are going to   check when X is equal to negative 1 well well  this is a quick check that we can do if you put   negative one into the function right here well  enough 1 minus 1 is enough 0 its undefined it's   bad so it's not going to converge for sure but do  not just rely on the function every single time   right do not rely on the function every single  time because if the function doesn't work yes   the power series won't work that's for sure but if  the function works it does not mean that the power   series will work right so I will still show you  put a negative 1 right here so that's just to get   this the sum as n goes from 0 to infinity we have  negative 1 to the nth power over n plus 1 and put   the negative 1 here we have negative 1 raised to  the n plus 1 power like so and then just simplify   this real quick this is the sum as n goes from  0 to infinity negative 1 to the nth power over   n plus 1 well well this right here is what this  is negative 1 to the nth power times negative 1   to the first power right good now check this out  this part negative 1 to the N times negative 1   to the N is negative 1 to the 2n so let me just  register this is negative 1 to the 2 it means   it's always 1 so they pretty much cancel off we  just always get 1 but we have the negative right   here though negative 1 to the first power so  we have this negative in the front and then we   have the sum and goes from 0 to infinity again  this is just going to give us a 1 and over plus   1 like this now does this converge no it does not  because if you're ready out real quick this right   here is actually well you don't need to write  a you can do the trick that I showed you guys   earlier they mean two doubles you gotta get this  right here is the sum let me minus 1 to the N so   I get 1 over and only because n plus 1 minus 1  you get and we're to balance it we will have to   add 1 here and here so we get n is equal to 1 to  M plus 2 infinity plus 1 to infinity notice this   took better yeah this is the P series when n is  equal to 1 what also say this is the harmonic   series well this right here I'll just call this  to be the P series you have just use that piece   one which is this or equal to one like this well  of course it approaches and when you have negative   x this actually goes to infinity so you actually  get negative infinity but doesn't matter right   here do not include negative one again you could  have to say because the function doesn't work of   course the series doesn't work right in that case  but let's check the next one check when X is equal   to or two well does it converge when X is equal to  I put the tool right here I get L and off-street   it works but no because this is outside of the  interval of convergence already right so to nap   just use the function right do not just use the  function I will put a 1 not only in the function   but actually right here right so be sure you  really use the series all right put a 1 here so   we're looking at this one is the series as n goes  from 0 to infinity and this is negative 1 to the   nth power over n plus 1 times 1 raised to the n  plus 1 power the good thing is that this is always   1 so that's good so we have the series as n goes  from 0 to infinity put this on the side because   I like that and then right here we have the 1  over n plus 1 like this in fact this right here   we can the alternating series right because well  this so I'll call this to be the BN to check the   first check I don't know what I put on to first  first check the spean go to 0 point and goes to   infinity so let's go ahead and say s and goes to  infinity P n which is 1 over n plus 1 does this   go to 0 ensured us because if you put infinity on  the bottom one over infinity it's clear that goes   to 0 so the first check checks secondly do we have  p.m. plus 1 less than or equal to BN namely does   PN decrease well if you put PN plus 1 and plus  1 here we get 1 over n plus 1 again for this   expression just enter and plus 1 to this ad so  we have that but don't forgive is to have a plus   1 after this and this right here is it is this  less than or equal to the original BN namely 1   over n plus 1 well you can cross multiply if  you'e like this times that is just n plus 1   is this less than or equal to where these times  does just n plus 2 and this is so true isn't it   so checks therefore this right here count verges  this series right here converges let me actually   put this down here converges by the alternating  series test and perhaps this is you know what   you have to show I know it's a long question  anyway gradually converges so you come here   you put down a bracket that means you include the  endpoint right here this is why I told you guys   that you really have to check the convergence at  the endpoints when you do calculus with the best   fret which is integration which is the calculate  which is the calculus step so be sure you do this   okay so box this for the answer so take a look  take a look take a look okay so yeah continue   continue continue we just have 20,000 more to go  you shouldn't be taking two though I don't know what's the next question let's see what's the next  question in first tangent all my guides it's going   to be another long one anyway number seven here  we have the inverse tangent of X and the centre   is at zero well well what's the connection  between in first engine and our best for maybe   well it doesn't have like direct connection but  it's like a long distance relative you will see   because we know that if we differentiate in first  tension we get what 1 over 1 plus x squared so if   we integrate 1 over 1 plus x squared we get in  first tangent so the strategy is I'll just put   on a note right here on the side for you guys  if we integrate 1 over 1 plus x squared this   will give us in first tangent of X all right  and yes I do know to put a plus C all right so   we'll just come over to series for the 1 over 1  plus x squared and integrate that yes all right   so go ahead perhaps I'll just write on the best  friend again 1 over 1 minus X this right here is   the sum as n goes from 0 to infinity X to the nth  power absolute value of x is less than 1 and now   1 over 1 plus x squared this is the same as saying  1 over 1 minus parentheses negative x square like   this ok and then I'm just going to enter this  right here so we have the sum as n goes from   0 to infinity parentheses that and we have the  negative x squared to the nth power enter this   into the absolute value absolute value of negative  x squared is just 1 good now work this out you see   that this is of course 1 over 1 plus x squared and  simplify this a little bit this is the series as n   goes from zero to infinity we have negative 1  to the nth power so I'll put this down nicely   and then this is X to the 2n power like this  and of course when you have the absolute value   of negative it doesn't matter and then you can  take the square root equity earlier so oh no we   get the absolute value of x is that stuff 1 thank  you so just some simple algebra for that now it's   time for us to integrate we will just integrate  integrate left hand side in first tangent so here   we have the inverse tangent right hand side first  of all put a c-plus unfortunately we can now say   plus C in this case C plus and here don't worry  go ahead and I want to the power this is the new   power divided by that man we can integrate power  series all day all night long right ok so here we   have the sum as n goes from 0 to infinity again  when you integrate you don't lose any turn right   when you differentiate maybe sometimes you want  to stop it anything for the 1 et cetera anyway   here we have negative 1 to the nth power over 2m  plus 1 and then this is X to the 2n plus 1 power   like this and yes you know it we have to find out  what the C is and yes you know it we are going to   pick x equals 0 and then put it here we get to  use the purple pen now this right here is the   inverse tangent of 0 and of course that's going  to be 0 no PD Oh put the C Class series and goes   from 0 to infinity negative 1 to the N over 2  n plus 1 you put a 0 in here so you have 0 to   the 2n plus 1 power this right here is equal to C  plus well when n is equal to 0 the first thing is   to do to first power times all that which is 0  likewise the next time point is 1 get 0 and so   on so and so on so you get zero for this series  in other words C is of course equal to zero that   means right here it's equal to zero okay so we're  going to to poverty earlier so just kind of stay   strong okay so let me just write this down here  we have the inverse tangent of X which is that   we have the sum as n goes from 0 to infinity and  we have negative 1 to the nth power over 2m plus   1 and we have X to the 2n plus 1 power well the  radius of convergence will stay the same because   we just integrate this thing right here right  so R will still be 1 and that means I will be   going from negative 1 to 1 but this time we don't  know how happen at the endpoints so go ahead and   do the check here we go this is the check when  X is equal to negative 1 again do not just put   into a function ok you're always had to put in the  series and just observe unless you can say ok the   function doesn't work the series doesn't work  but you know in this case in first engine you   always work so don't do that so put this right  here so here we have the series and goes from   0 to infinity and this is negative 1 to the nth  power over 2m plus 1 and here we have parentheses   negative 1 to the 2 n plus 1 power like this ok  here what do we have this is the sum and goes   from zero to infinity this is negative 1 to the N  over 2m plus 1 and of course we can separate this   this is the same as negative 1 to the 2n times  negative 1 to the 1st but this right here is nice   to equal to just 1 this is the negative 1 I can  put it all the way in the front this is the sum   I'm goes from 0 to infinity so we have negative 1  to the nth power and this is 1 over 2n plus 1 like   this right so that's how you do it then does this  count virtual diverge what kind of series is this okay good so take this to be the PN and that's  good to go ahead into the check first check   thus PN go to zero so as n goes to infinity P  n which is 1 over 2 plus 1 to infinity to here   that's equal to 0 yes it does so of course checks  secondly do we have a decreasing PN where we don't   know yes so let's put this down to n plus 1 here  so notice 21 here okay 1 over 2 times n plus 1   and then plus 1 is this less than or equal to the  original PN well just work this out real quick so   here you get just cross-multiply if you like  I find the students like to cross multiply so   this times that is 2 and plus 1 is this less than  what you called whoo this right here is 2 n plus   2 plus 1 so it's of course plus 3 and if you look  at this of course each X so again this right here   let me just read it down here you can't verges  by a T okay so you can't purchase by AST okay   so we can come here and include negative 1 very  good now don't forget to check when X is equal to   positive 1 yeah anyway put a 1 in here so we are  looking at the sum as n goes from 0 to infinity   and here we have negative 1 to the nth power  over 2m plus 1 times 1 raised to the 2n plus   1 power the cool thing is that this is always 1  yeah so we're looking at the sum as n goes from   0 to infinity I will just read yes negative 1  to the nth power and then this is multiplied by   1 over 2m plus 1 like this yes this is the second  the same is that part this Reggie has a negative   is a negative finite value this is just a pass  to you so I will just tell you this right here   converges as well of course because we can do the  same thing converges by AST okay the airport will   come here and include positive one as well that so  well done ladies and gentlemen this is the inverse   tangent of X take a look I know another monster  question but this is what we have to go through   in calc - okay so have a look in the meantime let  me tell you guys why I'm drinking this theme okay   just black key okay now of course we are going to  continue with the next question we have a couple   more to go this is an hour already man several  questions took me 1 hour oh my god it's at 55   minutes so almost an hour yeah hopefully Tony will  be better and yeah I repeat some questions that I   know you guys will have to know for your childhood  classes and the questions that typical questions   and I think a copper pretty much everything  good enough so now we'll be talking about the   best spread at a different Center so here we go  this is number eight I'll kind of walk this out   carefully with you guys I had a equals three  okay so this is what we are going to do let me   just write down the best friend okay I'm for you  guys right so works it sounded perfect but I will   just do this well still ready down for you guys  this is the original base rate 1 over 1 minus X   the original pair spread this is the sum as n goes  from 0 to infinity X to the nth power absolute Phi   of X has to be less than 1 okay the original best  friend now this is our best friend but he moved   to a different center so this is what we do start  with 1 over 1 minus X that's exactly the one that   were doing but here is the deal here's the deal if  the center a is equal to 3 then what you want to   have is this I still have a 1 on the top that's  ok but if you have the a right here is 3 what   you want to have is you want to have X minus 3  as you were input they have to be together why   because remember for power series it's going to  be the sum of the coefficients times X minus a   right to the nth power so like this right so you  want to have this as your input that's the idea   now we are going to make sure this is the same as  that algebraically of course so now if you look   at this we have X plus and well X minus 3 that's  no good because originally I have a 1 mm that's   X minus 3 and by the way I want to maintain a  minus so let me actually put on a minus right   here so technically right here I actually have  negative x plus 3 isn't it in that case how can   I get a 1 back we oh we can just put down a  minus 2 like that again this is just algebra   word that you have to do and remember if you have  X is equal to Streif if you have a is equal to 2   for the center you must have X minus 3 as your  input and they turn you'll see I mean by that   so always put this down right away and shows  just try to see if you have to add or subtract   so you can match the original and again I put on  the minus here because we want to subtract this   right here okay so that's the idea I'm going to  erase this now and then we do some what algebra   here one is of one on top of course over but I  don't want a negative two I want to have a 1 so   what we can do is just factor things out so here  I will factor out a negative 1/2 like this I'm   gonna fight on negative right here as we okay and  then our open the parentheses originally I had a   negative 2 factorial right here that means I have  a positive 1 originally you had a - battle - it's   a plus but that's no good because I want to use  the best friend it's a - it's okay I'll look at   this s - - okay and originally we have the X  minus three in the parenthesis but we factor   the other two so be sure we divide this by two so  in fact this will be our input yeah crazy huh yeah   when the batsman moves away it's crazier that's  okay don't worry we can always find it this right   here is negative one over two this right here you  enter this thing in here and in here so we have   the series as n goes from zero to infinity open a  parenthesis and should enter this in here negative   parenthesis X minus three and then over two and  then raised to nth power and then absolute value   that and then open this right put in here and then  yeah and then it's always there now of course all   right and then just work this out this is negative  one half no big deal yeah perhaps or still keep   down great why not all right to continue this is  a series as n goes fast zero to infinity now this   is probably to do negative 1 to the nth power  they made ready down first and then through to   the nth power maybe you guys can keep track of how  many and then say I'm saying right now so we have   negative 1 to the N and then 2 to the nth power  and we will have X minus 3 to the N power so as   you can see this is for me a coefficient times  X minus the center which is the a is 3 and then   to the nth power like so finally this is negative  1 I'm just going to group this and multiply with   that so perhaps I've already down here can we see  okay and is equal to 0 to infinity yeah kind of   you see the summers and goes fast you rotary  energy okay good okay negative 1 times negative   1 to the end of course we just add the exponents  so this is negative 1 to the n plus 1 power then   2 times negative so require methane 2 times 2 to  the nth power we will just divide this by 2 to   the n plus 1 power like this then this right here  stays x minus 3 to the nth power like so very good   huh now we just have to work this out absolute  value of the negative doesn't matter multiply   the 2 on both sides so we see absolute value of X  minus 3 is less than 2 that means the R is 2 well   where's the ident this time the center is the  street so let me just draw a little picture for   you and what we do is move to the left twice  so if you just move to a lab twice you get   one right here right and then if you move to the  right twice you get 5 so minus 200 plus 2 so we   have one two five now notice what did we do with  our pair spread just 10 equals the algebra right   there for you do not include the end points for  the eye just at the original best friend okay so   this right here is the answer for the best friend  if it moves to this new center 3 that's it okay perhaps I need more space for next one so I'm  going to erase that there is this question okay now we are looking at one over X square  when the center is at negative two and braids   no that was number eight this is number nine yeah  okay number nine here we have 1 over X square and   we have the center at equals negative 2 you have  two ways to do this the first way is you can still   use your best use our best friend and the second  way is that you can actually use the father of   the power series namely Taylor yeah you can use  the Taylor formula together but I will show you   guys how we can use our best friend for this hmm  well for the best friend is X to the first power   on about huh this is x squared on the bottom but  we know if we differentiate best from we can get   the x squared something on that on the bottom  so we'll do that but the input here is just X   let's see how we can do it hmm well this is how  welcome do it first of all first of all we need   a power series this is just the power series  of 1 over X I want to differentiate this guy   later on I want to look at this guy I want this  let me just let me just spray this down for you   guys this is what we want first we want 1 over x  add a equals negative 2 again why because we want   to just differentiate this so you can get that  and it's going to be pretty fast okay so let's   see how we can make this happen again here is our  best price so let me just write this down 1 over   1 minus X this is some dedications seriously I  write on the best friend on the how many times   already that will continue right down the path  spread for everybody so this is our best friend   right here now write down the things that you  want to do which is this okay so write down 1   over X like so this time a is at negative 2  we want a 2p and negative 2 that means what   that means I need to look at 1 over I need the  input here to be X minus negative 2 which is X   plus 2 like this hmm okay well this and that  are totally different but it's not bad at Oh   because first of all well I can put down down  as a plus ya know just put it on purple and we can just - - right here yes very good isn't  it again if you work this out you just kill the   X back yeah so that's what we have then let's  see I will just do one more algebra I will just   do the ultra here okay to use the best friend  in water first use the best phone we will use   in water first use the pair spread we need to  have a 1 right here right so this right here   I'll just read yes 1 over I will factor out the  negative 2 right there so I will write this down   s1 and then negative 1 over 2 like this ok times  1 over parenthesis here we go original you had   negative 2 we factor the out so we just have a 1  then originally were positive but we factor they   are negative so this is minus which is much more  prefer tea anyway right and this is what we had   originally but we factor the other two negative  210 yadi and then we'll just have parenthesis X   plus 2 divided by 2 like so yep then we can use  our best friend this right here is negative 1/2   this right here is the sum as n goes from 0  to infinity we just enter all this right here   in here and here so that's going to that so we  see that this is going to be parenthesis X plus   2 over 2 raised to the nth power like this and  perhaps I'll just ready down a little bit more   here we have negative 1 over 2 and this the sump  as n goes from 0 to infinity and let me write it   down like 1 over 2 to the nth power and this  right here is parenthesis X plus 2 to the nth   power like this and finally I am going to just  write this down for you guys I'm just going to   multiply the negative in and then distribute  this here so we get the sum as n goes from 0 to   infinity and we have negative 1 so it's negative  1 over 2 to the n plus 1 power and then we have   this which is X plus 2 to the nth power like this  right I know you guys might not be able to see I   don't worry I will already down over there again  anyway now what I ready done yeah I don't know okay so I'm just going to ready down with you  anyway what this is doing is the following we   know that 1 over X is going to be this we have  the series and goes from 0 to infinity again do   the algebra hopf doing piling race for no reason  here we have the negative 1 over 2 to the n plus   1 power right here so I'll just put that down  and that's the coefficient and then here we have   the X plus 2 raised to the nth power like so okay  and don't forget to enter this right here in the   absolute value as well so to the right here real  quick absolute value and X plus 2 over 2 is less   than 1 so again multiply both sides by 2 worldcat  the answer for that anyway this is not a big deal   the main deal is that we were different shape  both sides so DDX I'll just put on T DX T DX okay so differentiate the difference I'd this  right here just radius X 4 negative 1 power   bring the negative 1 to the front and you get  negative and you get extra negative 2 power so   you get 1 over X to 2 power like this right  hand side man I love differentiating power   series very very fun you'll see oh by the way  okay here we go first of all you bring the power   to the front and then minus one and get that  the Chen dude right here it's always going to   be one because remember it's always going to be  X minus a right so the Chen who is just always   going to be one so if you forget about the Chen  do you know you're safe this time anyway bring   the one to the front this time this is just a  negative okay and then over this is 2 to the M   plus ones power and then we have the X plus 2  raised to the minus ones power but now if you   put n is equal to 0 and here the first time will  be 0 so you can skip that you can just do it and   is equal to 1 and let me just double-check  my answer key you had that oh let's see oh yeah so now this is the person I have  this is not the first time upon the and so   keep out or you guess why I mean find the one  I found on the enter key anyway nothing yet   this is negative that's naked here of course  we can just multiple multiply both sides by   negative so just multiply and multiply both  sides by negative however you want to write   it yeah so that is an action so man I was just  ready some one more time so this is one over   X square and this is equal to next time this  times that cancel so you have the sum n goes   from 1 to infinity and over 2 to the n plus 1  times X plus 2 raised to the minus ones power   like this okay then let's see where the  me just double check my answer key most   likely the one doing Scott mines the kiss knock  right so that's the one I'm not trying to fix let me just double-check triple-check yeah so I  will just triple check the in circulator right   here yeah anyway though right here just  walk this out real quick so you get the   absolute value multiply 2 on both sides or x  plus 2 is less than 2 so you know right away   or is to now the I the center this time is 2 so  let me draw a picture for you guys real quick   all the center is a negative 2 my back then  center is negative 2 yeah and you go to the   left twice which is going to be negative what  and go to the right twice which is zero so you   actually get negative 4 comma 0 like so yeah  okay and what you are going to do is hmm well   again you should plug in negative 4 in here  to see if we can purchase for now but if you   plug in negative 4 here then you will see the  massif check negative format let's see that see I just want to make sure what I do that I think I got this answer from wolf on  alpha or so then that's why I'm like hmm let me just double-check on one offer real  quick and nothing wrong with this I just don't one over X square yet not too oh so I do have a typo on my in psyche so our  face stuff for sure I don't know why I'm not   saying ice five times already but yeah so I'm wolf  on alpha you can see that's the answer maybe you   guys cannot see but it's just that they fixed  index so sometimes it's annoying but this is   okay all right so what you're going to do next  is just go ahead and do a quick check when X is   negative four check this okay so you get the sum  this is when X is equal to negative okay just go   ahead and do that real quick n is equal to one  to infinity and you have an over two to the M   plus one and when you put negative falling there  you get negative four plus two and you have the   M minus ones power all right and what to do is  this this right here is going to be the sum and   goes from one to infinity this is n over two to  the N times two to the first this right here is   going to be negative two so I will have negative  one to the n plus -1 power times and if you do   negative T one already so this would be two and  then you have the M times two to the negative   one you see this and now cancel this right  here diverges because similar to what we did   in a previous video in a previous question so this  right here it's parenthesis if you have zero here   it's going to be a similar situation right so you  will have this and that will be parting you and   that will not be convergent so just use the test  for diverges right so the limit does exist yeah   anyway I will leave this to you guys for that part  ok so that was number nine took some time for that now if it's a typo i'ma answer key and yeah and  let's see okay for the next one mm-hm let's see   that's still kind of like best friend you'll see  quite oh this is number 10 we have one over and   here we have a quadratic again but this time three  times x squared plus 6x plus 10 at the center   negative three okay what do we do can we factor it  unfortunately not can we complete the square yes   we can so let's go ahead and do that so perhaps I  will just put on the completing square on the side   for you guys real quick if you look at x squared  plus 6x and let's leave a space let me just put   on plus 10 right here because for this we will  have to use the magic number to do so you look   at this number be sure you have the 1 in front  of the x squared which we do so take 1/2 times   the coefficient of x and then square that this  right here is 3 squared which is 9 so go ahead   and just add nine but we'll have two - a night so  that you don't change anything the left hand side   will not leave X on this side with these three  terms is just going to be X plus three squared   if you complete a square if you factor you get X  plus 3 square and this and that it's going to be   plus one so this is the irreducible quadratic but  it's actually pretty good you see why this right   here on the bottom now is 1 over I will just put  down one goes first ok Plus parentheses x plus 3   squared like so here it's a really ridiculous and  like the question start with the earlier you have   to kind of get a queer stuff right you see the  input here is already X plus 3 when you have the   a is at negative 3 that means you have to have  X minus negative 3 namely the X plus 3 which we   do already and yes you might be wondering what  if you have this and you want the center to be   a zero where you want the center to be at like  five I will tell you the question might be much   more difficult than this but that's not a plan  for today so maybe next time or so all right   so to do this now what we want is the following  again use our best friends to our cousins Kirsten   so anyway put down the best round one more time 1  over 1 minus X this right here it's equal to the   sum as n goes from 0 to infinity X to the nth  power absolute value of x is less than 1 well   if you look at the big picture this is what we  want we want 1 over 1 plus x squared here which   is actually not so fat what you make a noise I  don't know this is actually not so bad this is 1   over 1 minus negative x squared we did a few times  already and I can just put this right here in here   here and all that so this is the sum and go spawn  0 to infinity parenthesis that enter negative   x squared to the nth power put this in the up to  value as well right and if you look at this we get   1 over 1 plus x squared like 1 plus 1 x squared  1 over 1 plus x squared this right here is nice   to equal to this we have the sum and goes from  0 to infinity negative 1 to the nth power times   X to the 2n power like so negative doesn't matter  take the screws on both sides and you end up with   up to 5x it's less than 1 so this is actually  the one that we are gonna use this time okay   because now we have this we have 1 over and we  have this right here which is 1 plus parenthesis   X plus M just write it down we have the 1 over  1 plus parenthesis X plus 3 and then square I   so right all we had to do is enter X plus 3 in  here in here in here what done and in fact you   could have enter oh you could have used the best  friends curse of 1 over 1 plus X yeah man I should   have done that okay let's see let's see that see  that see that see that see that see that see yeah   you could have just used 1 over 1 plus X and let's  see if I want to do though for you guys or not I know that see that see I don't know complicating  so I will show you guys the quickest way to do   the power series for all this so I will look at  this I will look at this and perhaps that's just   make the time to them right here okay so 121 so  let's see hmm well if you have this you can just   look at 1 over let's do 1 over 1 plus X which is  1 over 1 minus negative x I know it's going to   be real symbol about you know this right here  it's actually no no no what am i doing series   no I'm just doing one step right maybe I'll just  make a cut but yeah save one take but this I will   not make Cavanaugh anyway one now this is what  we have so we have 1 over 1 plus parenthesis X   plus 3 squared and now in what if to use our best  friend okay we have the one on the top this is   the one that's good but this is a plus so we do -  so we do minus minus parenthesis X plus 3 square   like this right so what we can do is of course  enter this into here and that's it much easier   so we have the sum and goes from 0 to infinity  enter this right here so we have a parenthesis   with negative parentheses x plus 3 square and  to the nth power and don't forget to enter this   into the absolute pocket as well so we have the  absolute value negative x plus 3 that's step one yeah and then work this out so finally we'll  get the sum as n goes from zero to infinity   this is negative one see this is much faster I  don't know why today I'll use negative 1 to the   nth power and then this is X plus 3 to the 12th  power like this and we're done and of course this   is nice apps the five negative doesn't matter so  we have absolute value of x plus 3 is less than   1 and this is absolute 5x plus 3 is less than 1  so this will mean R is equal to 1 R is equal to   1 and the in kpop convergence well don't forget  we start at 3 we start at negative 3 move to the   left one time which is negative 4 and move to the  right one time which is going to be at negative 2   so we go from negative 4 to naked YouTube and  we do not include the endpoints because again   which is the algebra which the best friends and  it's cousins all that stuff alright so this right   here is it okay so let me just double check yeah  just like that it's actually pretty fast I don't   know I just come with the cousin earlier but  you don't have to you can just do this right   here okay so that was 10 questions and took me  84 minutes to explain all this hopefully you   guys are finding this helpful okay and if you  guys do please just let me know no way I would   know what kind of materials I shall make for you  guys okay man number 11 yeah we go number 11 e to   the X sin 30 add a equals 0 e to the X centered  es 0 that looks like a face let me just write it   down 11 still looks like a face forever right  okay so this time unfortunately we cannot use   our best friend anymore we can integrate this we  cannot differentiate this to ke to X you know what   doing any algebra so no but it's okay because we  can use the Taylor formula right the father of   the power series I will say and this is what you  do first of all I will just write down this way   you guys will make a table set this up we have  to have ad and then the nth derivative of the   function and then the third part right here you  great on the telephone law which is the thing you   have to remember this is f the ends derivative F  and you enter in this case a is equal to zero okay   and then D PI D PI and factorial okay now let's go  ahead and do a few of this is your two enough as   long as you can see a pattern that's that's good  enough zero one two three and in fact you'll see   this is accurate enough first in first when n  is zero you don't do any derivative it's just   the original function so you write down e to the  X and this is just the original function and now   here we go differentiating e to the x yes we still  ke to the X that's perfect and then use two key to   the X and they still get e to the X and so on now  I'm going to enter zero into here so we have e to   the zero power over zero factorial and remember  three factorial by definition is equal to one so   this right here is what we have and of course  that's just to the rest we have e to the zeros   over one factorial and then again e to the zero  power over two factorial and then e to the zero   power over three factorial and if this is just  any end that you want of course you always can   eat the X and notice that e to the X is just equal  to one that's very nice so on the top it's always   one we can just put that out if this is an just  divided by n factorial that's all we have yeah   that's it so here we go let me just write this  down again and this is for the record any function   that you want as long as you can differentiate  that that means it's good enough you can have the   power series expansion thanks to Taylor you have  the following formula is the sum as it's a series   as n goes from zero to infinity CN times X minus  a to the nth power Taylor says the CN here is   exactly the one that with it the nth derivative of  F is already at the center a and then divided by   n factorial okay so I will tell you e to the X is  actually equal to the series s and goes from zero   to infinity the coefficient is nicely equal to one  over N factorial so I just put that out and in the   case right here is equal to zero so we just get  X to the nth power like so and that's pretty much   it and now you might be wondering how do we find  that radius of convergence good question well we   still want to have the absolute value of X minus a  to be less than R and let me tell you let me tell   you actually maybe change your color a little bit  we still won up to the power of X minus a to be   less than R and let me tell you this nice formula  for R this R okay what you do is go ahead and take   the limit as n goes to infinity and then apply  the absolute value okay look at the coefficient   sequence which is a Siad and this is pretty much  deriving from the ratio test now you don't do the   original research test if you want to use the  formula right here but they are what you do is   you do C and P by V by C M plus 1 in that just  come to that if the limit exists you get the R   for free well actually do the work so it's not for  free anyway that's what you have so this is what   we have then we're going to go ahead and do our  radius of convergence so here we go ours equal to   take the limit as n goes to infinity and sometimes  you can also say s and goes to infinity up to you   anyway here we go CN we'll choose this first  I will have 1 over n factorial right 1 over n   factorial that's this part and then notice we have  the 1 over C n plus 1 so we can just multiply this   by the reciprocal and M plus 1 means we have the n  plus 1 as the input and n factorial that and then   divided by 1 like this again this is the CN and  then when you have the CN plus 1 on the bottom you   can just flip that and you have the n plus 1 right  here factorial so that's what you simplify the big   factorial bigger factorial which is n plus 1 I'm  just ready as n plus 1 times the next is going to   be N and the next one is going to be a minus 1  and n minus 2 and so on so you can just improve   that and say that n factorial very nice because  this then now will cancel and you're looking at   the limit as n goes to infinity and this is just  M plus 1 we will take in and goes to infinity this   is of course equals infinity Congrats this is the  first one that we have today are equal to infinity   why do I say congrats because if R is equal to  infinity I is automatically negative infinity to   plus infinity doesn't even matter where the center  is all the way all the way so ladies and gentlemen   this is e to the X as a power series I am just  checked any messages I saw some message sorry yeah all right so this is e to the X and we'll  be using a same thing for the other ones and of   course you should know the e to the X by heart all  right this is another friend that you have to know   it's a standard function that you definitely have  to know now for the next one we have sine X so   here we go number 12 number 12 sang X add a equal  to zero man this power series business the twelve   questions took you an hour 30 minutes so you've  multiplied up by 10 that would be equivalent to   two like 100 differential equation and seriously  so I don't know if I show ever to that bunch of   differential equation anyway I will and then do  the work alright so again for science we will be   using the Taylor Puma to help us out but you know  in to differentiate the sign you care like cosine   of that and especially when a su 0 you get some  zeros so instead using and I'll be using Kate and   you guys will see why later ok so let me just  use K and right here I will be writing down in   case derivative of the function just differentiate  that however many times that we need to and then   here we have the K and then enter a is equal to  zero and then divided by n factorial divided by K   factorial all right and again like K and they are  the index that just a dummy variable so doesn't   matter that much all right so let's see I will  do a few of them I'll show you guys real quick   go ahead 0 1 2 3 4 5 6 first we guest annex and  here we go this is that the rotate derivative   fun so differentiate design X we get cosine X and  a do yet we can negative sine X and then do you   again we get negative cosine X and DUI can we get  past this ax and then do again we get cosine X and   then to you again you see that start repeats  yeah and so on so on so now putting 0 in here   divided by Y I produce Turkish 0 so that's gone  putting 0 for cosine we get past the 1 so it does   matter so we have 1 and then over 1 factorial so  let's write that down next when we put 0 in here   when we have sine gives you rope well next one  when you push through a ratio we can negative 1   divided by 3 factorial next pulling through into  a sign we get 0 and next we get positive 1 again   and that's over 5 factorial and the next one if  you're putting 0 you get 0 and if you would like   to see one more sevens it's going to be negative  cosine X and you can guess that you was going to   be negative 1 over sin if I told you anyway and  so on right so here's the deal you see that your   0 and this 0 does road a slowrun what you want to  do is do not try to write a formula with the 0s be   included the aesthetic overkill just don't worry  about it try to focus on a formula I try to focus   on a formula with this this this and that only  right so that would be the the coefficients that   we really care about well when K is 1 we get this  on K is 3 is quickly that in case that would get   that right but here's the trouble though we cannot  do not just say over K factorial here because if   you put an over K factorial if you do get the K  goes from 0 to infinity if you put this down you   will be including these guys and that's not good  so seriously that's not good so again do not look   at the K right here this right here is just forced  to sit at the table right I shouldn't have circled   the left-hand side I just noticed that but if you  look at this and now we will be using n so let me   show you what we'll do with it here is sine X I'm  going to write down the terms for you guys first   right for the first time this is the expanded  version this is what we have we have 1 over 1   factorial times X minus 0 so it's just times  X by itself right here to the first power ok   that's ok next we have this which is minus 1 over  3 factorial and we have X to the third power ok   and then next we have that which is plus 1 over 5  factorial X to the fifth power next we have minus   1 over 7 factorial X to the seventh power and so  on so on and so on like so remember remember the   formula is this and then times X minus a to the K  power right that's the formula for that so if you   have one that match that match damaged that's  that's okay and so that's that's ok right but   now here is what we are going to do we are going  to be writing a formula down right here with and   I would add which is equal to 0 to infinity again  we're just going to look at these things and then   write a formula with the Sigma notation for this  right one more time this and I match this and I'm   actually said not much because of the formula  of the CN times X minus data formula like the   K ok so it's that one okay all right anyway so  n star 0 hmm okay first of all the alternating   so as you have negative 1 to some power because  n star 0 and this is positive so to the nth power   works right because this is when n is equal  to 1 and you can negative so that works next   on the bottom here if you focus on one three five  seven let's just see our numbers well our numbers   is to add either plus one minus one depends on  your starting value because n star with zero it   would be plus one again you see 1 2 3 3 to 5 5  to 7 the difference is to every single time so   that's a coefficient of n because n star 0 so you  have to add 1 so so you get 2 n plus 1 what these   numbers and then you factorial that alright then  you see this is X and again this and that match   so this will be kind of power right here so this  is 2 and plus 1 power so that's it okay so this   is science and again the reason I use K over  there it's because we had the zeros in between   we don't want to write a formula with the swoopy  included because that will be redundant that's   why I use K on purpose and now just write down  the remaining terms the terms the actual terms   will tell the zeros and we look at the remaining  terms and write them down into the Sigma notation   and finally I use then so that's idea alright here  is the C yet so I will compute all real quick so   here we go R is equal to the limit as n goes to  infinity and again I was just ready stunt just to   emphasize this CN times 1 over cm plus 1 right  so let me just emphasize if I didn't emphasize   a career earlier CN is all this so I will just  write down negative 1 to the nth power over 2m   plus 1 factorial that and for the CM plus 1 I'll  just put M plus 1 in here and to the reciprocal   remember it's the reciprocal right so I was just  put in past one year so we will have to and then   M plus one like this in parenthesis and then plus  1 and then factorial that and divided by negative   1 and I have the MSDN plus 1 so this register  PN plus 1 as well like this cause that now I   will work this out in detail for you guys this is  the CM plus 1 this right here 2n plus 2 right and   a plus 1 so you have 2n plus 3 factorial that  so if you expand this you get 2n plus 3 that's   the first term and then next term is 2n plus 2  because you just minus 1 and then the next time   is 2n plus 1 and then 2 1 and then 2 and minus  1 and so on but you can just factorial this guy   and you stop the good thing is that this and that  can be cancelled it which is very nice and now we   are looking at the limit as n goes to infinity and  we just have this and by the way the absolute for   you right just pretty much make all the negative  past here so this and that don't matter so we get   half the value of 2 plus 3 2 plus 2 yes it will be  infinity when n goes to infinity so this is just   as nice we get R equals infinity likewise I will  be negative infinity to positive infinity as well   do now include infinities that's why I just put  down parenthesis right away okay so it's just like   this one cool now I will show you how we can find  out the next one within this little box right here   ok so the next one number 13 we will be dealing  with cosine right so for cosine I will just put   this down right here cause I and again centered  at a equals zero don't worry I will be doing with   difference in theirs in the next few questions  I saw just hold up to that and let me see alright so let's see if there's any connection  between sine cosine yes yes there are a lot of   connections that the homies but the best  connections of course if we differentiate   side we get cosine and that's very nice so I  will just go ahead and do this for you gasp we   know cosine X this right here is equal to we  can just appreciate I'll put this down we'll   just differentiate Thanks that's pretty much it  right and of course we'll be looking at the power   series so let's go ahead and say differentiating  sine is all this so we have the series as n goes   from zero to infinity and we have negative 1 to  the nth power over 2n plus 1 factorial that and   then X to the 2n plus 1 power like so right and  do not put on the X this is for differentiation   now check this out take that the root of this  go ahead bring this to the front and then minus   1 well when you bring this to the front you  have the 2n plus 1 here isn't it this right   here it's a same as 2 n plus 1 times 2n times 2  and minus 1 and so on in another word to add in   your parentheses factorial that best thing is  this and our Kenzo nicely so I will read this   down for you guys this right here we get the  series as n goes from well notice this and   that cancel if you could see roll into the end  the first 10 stays this nostro turn so and will   still go with 0 to infinity this right here is  negative 1 to the nth power over this is just   2 and factorial now and here we will just have  X to the two ends power and again and the earth   star with zero because if you push through right  here and right here zero factorial by definitions   1 so you do have to start with an equal zero  in this case this right here it's a series for   cosine and the cosines that are will stay the  same because we just differentiate the power   series and ours infinitely well I will be going  from negative infinity to positive infinity so   this is it this is what cosine X like this very  good hey man there's so many power series I don't   know if I can narrow this or not hmm all right  anyway this is 13 so we did have mirrors already now number 14 is the time that we are going to  kind of play around with the center so as usual   here we go number 14 years right here wait that  was number 14 12 13 I was 13 yeah so number 14   we have e to the 3x but this time the center is  going to be add to two choices one you can still   do the table and this time you have to plug in  2 and you can produce that I did that before   so if you want to check that I can check that out  here I'll show you get a different way write down   what you are working with which is e to the 3x  this is the one that you you are trying to get   right and you should also put down the one that  you know apart that's that's the leader this is   e to the 3x what you want to do is well if the  center is at 2 then that means you should have   X minus 2 as your input okay and because we have  this string the France of course don't forget we   have the street in a front leg so and of course  we have the e so we have e to that power ok and of   course if you distribute the power this is going  to be 3x times what this is this is going to be   negative 6 right this is of course not the same as  that you just have to make it the same thing would   be happy we need the negative 6 and this is e to  the negative sixth power but to cancel that so   it's just going to be multiplied by e to the past  d6 because if we have e to the positive 6 times e   to the negative 6 you just add the exponents  just kind of introduce new things that's it yeah that's it and now which one are we going to  use of course what use our other fret I don't know   which ones matter friend of course e to the X is  our other friend all right so this is the one now   we are going to use I will just put this down  right here I'll just say use e to the X this is   equal to the series as n goes from 0 to infinity  and again you just have to remember this one over   N factorial better yeah you can just derive it  real quickly all right anyway X to the nth power   this right here works and because ours infinity so  don't worry know why but you don't need absolute   don't even need that all right so here we go this  right here we have e to the 6 power that's great   this right here will be using that so what we do  is of course we have to stop and goes from 0 to   infinity and all we have to do is put everything  in here right that's how we get at that so that's   the idea and they just put everything here so open  the parentheses with 3 parentheses X minus 2 and   then raised to the nth power like this that's it  and then the best part is R is infinity and I is   negative infinity to infinity so you don't have  to worry about the endpoints there's no end points   anyway all right we're still kind of simplify  this a little bit oh so this is e to the sixth   power and this is the sum as n goes from zero  to infinity that's right yes 3 to the N and this   is X minus 2 to the nth power multiplied in but  there's nothing we can buy so I will just write   yes oh I forgot the over factorial it's my bad  I put this in here I forgot already as the over   factorial I got all right so three to the N and  this to the end and then over N factorial right   and finally we are just going to multiply this  in so we end up with the series as n goes from   0 to infinity and depends how you want to write  it already as e to the 6th power times 3 to the   N over N factorial I know this looks like super  intimidating pie it's actually not so bad right   for 2 and power and of course R is infinity e  is negative infinity to positive infinity and   we are all done like this hey Vanessa I realize  how to erase this every single time because like   I just want you to be like clear so I had to  write it pretty big in order and hopefully the   poor is not soldered it looks okay right here  named and let me know if you guys I study for   your calculus 2 midterm we'll find out what so  ever right so anyway number 15 mile 15 man let me   tell you you guys don't know how I also searched  this just leave my mouth and just do the work sine   X add a be negative PI over 2 like that ok ok  so here we go sides and again now this is the   most important part whatever you're looking for  got to write this down a is negative PI over 2 I have forgot already tally of which does matter  alright whenever you have the a and anything psi   zero be sure you write es X minus da which is  going to be plus in this case and the PI over   two so this would be you a new input that you want  so that you can use the app seems to have job okay   and we have the sine at PI over two wait maybe  some power to is plenty cell power to sources   or let me do it all right sign at a equals PI over  two right here we've got question every team okay   so here we go this is sine X well in this case we  won the a to be at PI over two this makes you want   to have a factor X minus PI over two I hopefully  we can do the some connections between driver now   here's a deal can I just putting the sign right  here is this true no unfortunately not yeah so   X is of course not the same as sine of X minus  PI over two unfortunately this is not correct   hmm what can we do though well don't worry if  you can on your side maybe custom we help us   out and the truth is cosine can really help us  out and let me just write this down right here   for you guys okay so just how to see how to use  the identities you know that to help us out so   I will just put this down right here if you're  looking at paper - well because sine cosine or   did the constants for cosine the constants for  Complementary if you have set up the angle well   cosine of the complementary engulfed the eggs  it's the same as dosa let me write it over me   so the idea is that cosine of theta this right  here is I'm saying sign of the complimentary   and go of that so remember complementary means  that two angles are to be 90 degrees but we're   all those now so we are to use PI over two well  this is Theta already that means the employee   right here I would need to have pi over 2 minus  theta right so this is the connection similarly   you can also write yes sine theta equals cosine  of PI over 2 minus theta like this so this is   the co identity that were pretty helped us out  right we're in this case we are looking a sign   right we're looking at sine theta sine X over  there so it looks like we have the cosine to   help us out well I need to have a factor of X  minus PI over 2 by the definition oh this is PI   over 2 minus X minus theta oh don't worry this is  cosine this is all this is actually pretty good   notice so notice if we have cosine of negative  angle thanks to cosine being an intern found   an even function this is the same as a cosine  of theta so what we can do is in fact this is   the same as saying cosine of we can switch this  and I'll just negate hat negative and I'll put   this down first theta minus PI over 2 right I'll  just do some algebra here but this negative can   be disappeared because cosines even so a cosine  of theta minus PI over 2 like this so sine X of   course I'm using data but the same as X this right  here it's in fact the same as cosine of X minus PI   over 2 so this is very very powerful I will say  very very convenient and now what we can do is   I will just show you we can now use cosine of X  as the power series which we did earlier the sum   as n goes from 0 to infinity negative 1 to the  nth power over 2 n factorial times X to the 2n   power like this ok and notice that we mentioned  that cosine its event and we have this property   and you see that in fact we have the even power in  the Taylor expansion or the power series expansion   for cosine that's why you know even if it when  you have even function you can even exponent in   the power series expansion very very nice huh  so this right here we're almost done okay you   can do a table but sometimes your certainty will  be much quicker so here we go all we have to do   is enter this here here right and by the way will  you know how could you sum it all just register R   is infinite anyway put this down here so we have  the sum goes from 0 to infinity and we have the   coefficient being negative 1 to the nth power over  to earn parentheses that factorial that and then   here is just X minus PI over 2 raised to the 2n  power very very convenient isn't it and that's it   that's it and very quick our is infinitely that  means I will be automatically negative infinity   to positive infinity done done deal right the  more we know the easier life will be sometimes   so that's idea and I wrote these dungeons for  you guys sometimes you might be looking for like   cosine like PI over 2 name maybe this right here  will be helpful but you had used sine ping and add   function to fix that okay so that's number 15 and  man this is exciting we are kind of almost done right just to show you the time okay right now  is 8:45 8:45 p.m. the date is wrong on my watch   so don't worry too much about that oh yeah next  one's number 16 that's pace a little bit number 16   again we're looking a sine of X but this time add  a equals negative PI man negative PI so you know   that yeah we abuse at nippy right here again so  let me just put this down we have sex hmm well a   is negative PI that means I need to have an X plus  PI factor okay hmm well let's take a look right   here let's say we are on the unit circle let's  say we have the point right here and of course   remember the sine is the Y value this is right  here it's a sine value okay sign all fiber sign up   the angles as a single theta well if you have an  angle but H plus PI that means just go 180 degrees   so just go right here directly down here and this  is actually a really nice reflection whatever this   is you just come here and this is going to be sine  of PI plus so theta plus PI but the promise that   it was passed here and now it's negative so just  a sneaky that now this is the sine of PI this is   a sine of theta plus PI but this is just going to  be the negative value of that that's it yeah so   very nice saya X is equal to sine of X plus PI but  it's the negative so it's a negative on you let me   just emphasize that so why because again they  have a negative sign and again this angle here   is I should put a negative in you have why not huh  yeah if exist I say PI over three and then you add   PI yeah yeah the wife I'll use the same patches  opposite side for this picture I'm not sure if   I should put negative don't know if it does make  sense or not bad no no no no what I mean is this   right here is this distance right and that's the  distance but the connection is sine theta is equal   to negative sine of theta plus pi and again what  I mean by that is if you just go on to 80 degrees   more which is PI radians this distance the white  distance is just the opposite of that so that's   why you have negative all right so that's the idea  okay so that's what we have then what we're going   to use I'll just write it down again doesn't if  we remember this one up we'll be using sine of   course so we have sine of theta well that's your  sine X of course for I didn't decided to use data   of course all right here we go because right here  this is x over the key inside we have the sum as n   goes from 0 to infinity and what we have next yes  it's alternating negative 1 to the N power over   and remember sine is an odd function so see the  odd numbers and 2 n plus 1 good and then factorial   that x to the 2n plus 1 power and the radius of  convergence is infinity in this case well again we   just enter this right here and that's pretty much  it so we have this this is negative and this is   going to be the sum as n goes from 0 to infinity  just write this down pretty much negative 1 to the   nth power over 2m plus 1 factorial this guy and  then enter this for the include X plus pi raised   to to M plus ones power so and if you allow you  can just say this is negative one multiplied with   this negative on to dance power so finally we can  simplify this to be the series and goes from 0 to   infinity negative 1 to the n plus 1 power over 2m  plus 1 factorial that man I know it's like it's   not so bad but just how to write the same thing  over and over it's worse than integral sometimes   right anyway R is infinity because you're just  plugging and of course do I is negative infinity   to positive infinity pretty nice huh this is not  box spa whatever got it yeah okay so far so good oh my gosh next we'll have the size grip oh my god  man this is number 17 we chose to all these crew   shirts yes that person is me sigh square X add  a is equal to zero remember integrals how do you   integrate sine squared yes we should use the power  reduction sewing here we do the same as well so   here we go sine square X this right here is the  same as 1/2 times 1 minus okay for sine square   into 2 power reductions - and you will have cosine  of 2x just how we do the integral and luckily we   just want to have a is equal to 0 so we don't have  to see that X minus I PI over 3 or whatsoever this   is always need very very nice and now we'll  just distribute this a little bit this is 1/2   and this is minus 1/2 cosine of 2x and to exist  our input now and now let's see how can do with   this this is 1/2 this is minus 1/2 and for cosine  well let's remember this I just do this in your   head this is the sum and goes from 0 to infinity  what's the coefficient we have negative 1 to the   nth power over to n factorial this yeah and then  the input goes here to X and then raised to the   10th power and again R is infinity you can just  write that down you don't even in the absolute   political star to infinity that's very nice now  simplify this a little bit again this is 1/2 minus yeah that's just ready darling this sum I goes  from 0 to infinity this records data negative 1 to   the N over 2 factorial we can write this as 2 to  the 2 n times X to the 2n like that okay now check   this out I'm going to appear this in a so I will  just multiply this and that together so it seems   that we have the 1/2 all the way in the front and  I'll be adding because I will be multiplying this   and add so first I will have the sum and goes  from 0 to infinity negative 1 times this it's   going to be negative 1 to the n plus 1 power okay  so that's the that part and then over 2 factorial   you should be also of the parentheses and notice  how this is divided by 2 so we just divide this   by 2 then we can just you know do 2 n minus 1 so  this is 2 to the 2 n minus 1 power like so 2 to   the 2 and minus 1 because they can you / - right  here so yeah so you also have to consider this so   it becomes a divided by 2 like that okay so that's  idea and you have X to the 2n power like this so   it seems okay right but in fact we can do a little  bit better because the do is that if you look at   this expansion when any sequence 0 that's do this  in your head okay that's do this in your head when   n is equal to 0 ok this is going to be negative  1 so what's negative when n is 0 we have a 2 to   the negative 1st power so a 2 to a negative first  power over 0 factorial this is when n is equal to   0 okay and then X to the 0 power so this is just  that so that means you get negative 1/2 this is   when n is equal to 0 but when n is equal 0 you  get is and outside you have 1/2 so in thousand   cancel this out so the first turn here is negative  1/2 that can be cancelled yeah what's that passing   one huh all together we can just write down the  sum as and starting with one right because the   first time will cancel with that and then you just  go to infinity and just write on the rest plus one   totally 2 minus 1 or over 2 factorial that and  then X to the 2 1 like this yeah and then of   course our is infinitely that means I is going to  be negative infinity to positive infinity man this   marker is almost done but I have another one by  the way the smoker isn't so bad if you guys watch   my 100 series video you guys know that this marker  home it's not the one I like okay so that was   number 17 whoo okay now take a look take a look  take a look right do you guys miss that Taylor   formula maybe I heard some people say mr. Taylor  for moi sometimes we have to do that now sometimes   we have to do that just like you are trying to do  something new that we have nothing before then we   have to do that if not then just try to make a  connection then that's how we can make things   happen okay number anything right here I'm pretty  we have cosine X hat you guys ready a is PI over 4 okay think about how you will actually do this hmm  well if you want to do what the earlier this is   Cossacks and maybe you can end up with some kind  of function with X minus PI over four is that true   can you just do like sine or cosine to help us  out I don't think so unfortunately so here we'll   go back to the telephone if you know how to use  identity to do this one please comment down below   because I would like to know that myself I didn't  know how to do it but it's okay because the father   of the power series can always back us up real  quick right so here we go because at PI over four   we don't get 0 so I'll just use the end doesn't  matter so we have n right here and then do the   the relatives and then the formula which is the s  derivative at PI over 4 then divided by factorial   decay all right all right let's see one two three  four five hopefully you see the pattern already   but zero one two three four five here we go  cosine is the original next we get negative   sine next weekend negative cosine next we get  past the sign and then we get cosine and then   we get negative sine so of course we see that  repeat so that's good now I will tell you when   we put PI over four into cosine what do we get one  over square root of two all right so I will just   write this down here on the side cosine of PI over  four this is 1 over square root of two and we are   all adults now we don't have to rationalize the  denominator and the beauty of this question is a   sign of power but for it's also one over square  root of two so this is a beauty for that anyway   so I will write this down when we put hyperforin  here we get 1 over square root of 2 I will just   put it down right here and 1 over square root of  2 and we have the factorial let me just put this   down great actually let me put on this numbers in  red so when we do that we get 1 over square root   of 2 times 1 over 0 factorial like this okay next  we get negative so this is negative 1 over square   root of 2 because sine of PI over 4 is also  1 over square root 2 times 1 over 1 factorial   and you just continue next you have negative so  this is negative and again 1 over square root of   everybody has one over school - that's pretty  nice times 1 over 2 factorial then negative 1   over square root of 2 times 1 over 3 factorial oh  no this is positive positive and then naked past   passed one of the four factorial now it's negative  1 over square root of 2 times 1 over 5 factorial   so it doesn't seem so bad but here is the trouble  here is the trouble positive negative negative   positive positive negative negative so it's a 2  negative here to party here in the to negative 2   party is this alternating unfortunately not this  is not alternating in the usual sense like this   against negative negative and positive positive  and negative negative and so on so on so on so   this factor worth of a revision of this might not  help salt well we had to think better than this I   will put this down purple notice everybody has  1 over square root 2 so that stepparent that's   no big deal and you can just write on 1 over  N factorial again no big deal but I want to   do it yet one in purple negative one in purple  negative one in purple and then 1 and then 1 and   then negative 1 right in fact let's look at these  numbers right this this this this and so on you   guys will see there's actually a really nice way  to do this it does repeat so it's a good idea to   visit the unit circle and by the way if you want  to check out my unit circle emerge you can just   go click on that ok so now we'll just put this  down right here here is the unit circle and we're   always talking about 1 over square root of 2 that  means the angle is PI over 4 or maybe right here   or maybe right here well maybe right here right  and depends I don't like using if you like to use   sine cosine or whatsoever in this case it's a good  idea to use cosine check this out this right here   if you look at cosine right here this is PI over  4 and this is PI over 4 so let me just write this   down for you guys real quick cosine of PI over 4  is positive 1 over square root 2 and the moment   that we reach to here so you just have to rotate  here the moment we reach here this is negative   cosine of power of work if you want to today like  that but you can also look at the S cosine of PI   over 4 and if you want to go from here to here  this is actually a 90-degree angle so just have   to add PI over 2 which is going to give you 5 PI  over 4 in sub cosine this is going to give you   negative 1 over square root of 2 isn't it and then  if you keep going another PI over 2 this is rotate   PI over 2 rotate over 2 because you have a whole  circle cut into 4 pieces so 2 pi divided by 4 is   part work to ice all that stuff this right here  it's going to be cosine of PI over 4 I'm looking   at the starting and then I just have to add PI  over 2 but twice right that's the idea and you   can imagine if I were to NEX this right here I'll  just make you like a different - sign that so just   add a power - one more time and notice this right  here does give us negative 1 over square root of   2 right and for this point this is going to be  cosine of starting at PI over 4 and you add the   PI over 2 3 times like this and you end up with  positive 1 over square root of 2 because right   here you're talking about past the x value and  the negative x value and still negative x value   in a party Cassie negative negative and so on so  for all the numbers in purple is just cosine of   PI over 4 plus PI over 2 times and very nice so  here we go we can write down a formula cosine x   equals the sum and goes from 0 to infinity let's  write this down in fact I include us 1 over square   root 2 because of nicety will have the cosine  I so that's why it's very nice anyway we have   cosine starting at PI over 4 again starting at PI  over 4 actually let me match the color cosine of   starting at PI over 4 and we are going to just  add PI over 2 and like this because I'll start   with 0 so if you push it over here think at this  and then when n is equal to 1 just jump to here   so this right here covers the coefficients right  here but we also have to look at the factorials   on the bottom no big deal we can just look at  this and divided by n factorial that's it right   and because this right here is a is at PI over  4 so this is X minus PI over 4 raised to the   nth power like this this nostro terms so this  is very very very very nice let's see huh ok I   am suspicious again I don't know why I'm my enter  key I put on to nth power let me just double check   I'm just double check for alpha because I want to  make sure that I give you guys the correct answer and this work can take a little break  too and so it's a win-win situation yeah I got this right here on Villanova okay my  answer key is wrong haha I had a twin so we'll   just change that so as I said um I don't know  why whenever I come with answer keys I just do   whatever anyway this is yet and if you were like  and just work just out what's the formula I told   you guys before but RS infinity likewise I  will also be going from negative infinity   to positive infinity and Kattan okay yeah  so technical you should just work about if   you have a factorial on the pattern it's  actually just infinity just worked out okay now I have three special functions to talk  about number nineteen nine number 19 oops my bad   number 19 here this is the sign with an H next  to it and you can call this to be the think of   X right but don't worry if you haven't seen this  before because that how you get the definition of   this this right here is nice T equal to e to the  X minus e to the negative x over two like this   we want to find a power series expansion for this  add a equals zero right and maybe you haven't seen   this before that's okay because you certainly have  seen this right so all we have to do is use the e   to the X to help us sell for the power series so  we'll just look at this and just pretty much do   the work well I'll just write it down right here  because we can write a 1/2 in the front so we can   look at this as e to the X and then this is minus  e and this is negative x right so now we have   the 1/2 all the way in the front e to the X just  remember that this is the sum as n goes from 0 to   infinity and you have 1 over N factorial and you  have X to the nth power like this and then minus   the sum as n goes from 0 to infinity and you have  1 over factorial L radius negative x to the nth   power like this and it seems okay but this is one  thing I may not see hold on on a tee hmm oh yeah   by the way was just write down R is infinity and  of course we know I put a ton idea it seems okay hmm now this is the bizarre part okay so it just  kind of a beer with me this right here alternating   this right here it's always pasty and because they  are of the same kind so some terms are going to   get canceled and sometimes are not going to get  cancelled out so we had to pursue its culture   and depends if you want to stay with n whatsoever  and let's do this let's do this and that's do this   one over two it's one number two from this is that  let's just do the expansion okay so for this one   that's ready out the first term if you're putting  sin wrong here and here you can want so put this   down in red in black one next turn you get plus  X to the first over 1 factorial so splat classics   next we get x squared over 2 factorial next you  get X cubed over 3 factorial and guess can see the   pattern the next one is plus X to the fourth over  4 factorial and so on so and so on right so that's   the first term that that's a first part right here  now you'll just put this down I'll put this down   here okay and then I will have minus now here is  a small part if you put 0 in here again you end up   with 1 so that's ok then if you put 1 here this is  going to give you negative 1 and negative 1 hmm so   it's a minus X to the first power over 1 factorial  so it's minus X and then when you put - it's past   the again so it's plus x squared over 2 factorial  and you guys can see the patterns alternating   version so we have minus X cubed over 3 factorial  and then plus X 4 over 4 factorial and next one   is minus and so on so on and so on all right so  that's pretty much the expansion if you're ready   out and now check this out we have 1 minus 1 we  know 1 minus 1 it's equal to 0 so go ahead and can   them out then that will put on everything purp  off on the arm this is one half times this is X   but minus negative X minus minus becomes plus Z  so we end up with 2x right and the next you see   this and that - so they cancel out and the next  you see this and that well we get two of them   so let me just ready as plus two X cubed over 3  factorial and next this and that what cancel out   and you were like you see that this and this will  maintain and that will happen to X to the 5 over   5 factorial and so 2 X to the 5 over 5 factorial  and so on and just keep adding them up right now   the good thing about the 1/2 in front is that  when you distribute all the tools will be gone   isn't it so if you look at all this we get X plus  X cubed over 3 factorial plus X to the 5 over 5   factorial plus data forever here is the video I  like to just show you guys the expansion for this   because the terms will cancel out and of course  you don't want to write as zeros out you want to   just read our formula for the remaining things  that's very similar to the original sign this   is the hyperbolic sine cosine choices if you work  this out if you put in the cyma notation it does   not alternate this is n going from 0 to infinity  again this is the change thanks notice this add   it's very different this end are putting input  okay put a little purple kite so I know this is   the index situation that's a horrible sometimes  but if you didn't like the conclusion you can   just change the orders to K if you didn't like  the end that I used like twice and inside Y it   looks different so just change this to K that's  pretty much what I was doing with the original   sign question earlier you might be confusing but  once you get used to this you can do it too so   now there's no debate the powder and water anyway  nothing is alternating this right here it's just   going to be the following we only have the odd  numbers the factorials of that so we have 1 over   2m plus 1 factorial and n times X to the 2n plus 1  just like this there's no alternative let's think   Jack's ladies and gentlemen and you can guess that  you if we run through the you actually don't need   to run through because you know from e to the X  you're not R is infinity likewise I is negative   infinity to positive infinity do not include  infinities here you go this is the answer for   a cinch X right very nice isn't it now I want  to number 20 because you might guessed it what   number 20 is yes it's the hyperbolic cosine wait  so let's look at gosh now if your name's Josh   shut out you but this is cos e to the X plus e to  the negative x over 2 and a equals 0 right I will   erase this a little bit all right over you guys  don't mind well is there any connection between   things and cosh in terms of calculus of course  there is profiteth but what cannot derivative is   this well the truth is cosh of X is indeed equal  to if we differentiate seen jacks this is correct   hey this is true what we can do is you can just  go ahead and do little rocky right here the one   half stays e to the X stays the derivative e to  the negative x stays but don't forget to multiply   by negative 1 because of the chandu so you have  a negative right here and now what we think and   here is the beauty though or just actually togas  the following and I've just put it down right here   if you differentiate cosh you get things as well  this non negative the row of the cosh is past this   inch and the relative sage is past if Akash  just go ahead and try that out if you would   like by the way my job is to do the power series  so we will be differentiating sync checks which   is that over there I will just write down the  sum and goes from 0 to infinity and we have 1   over 2 n plus 1 factorial this guy and X to the  2n plus 1 and we know R is infinity so just put   that on it here this right here it's equal to  well do neither of you bring this to the front   minus 1 so you're looking at 2m plus 1 break this  down this is 2m plus 1 times 2n and then so on so   on and so on so you can factor it is so this and  that we can so and in the end you end up with the   sum and goes from 0 to infinity positive 1 over 2n  factorial this guy and then this is just X to the   2n yes this is the cost and you took alternate at  all and similar the hardest infinity is negative   infinity to passing infinity and you can be happy  I can be happy too because we are done right so   ladies and gentlemen this is cosh and you might  be wondering how can we get into the original   tangent because there's no easy formula there  is a formula but there's no easy formula for the   original tangent that's why we only did a inverse  tangent and for the tangent if you really want to   get a power series for that you would you can do  long division with sine over cosine what you can   do the Taylor formula so up to you now we have a  couple more to go point two more to go so number   21 in verse 10 the number 21 this right here is  the hyperbolic tangent but it's the Inc first   version of that I put a negative 1 right here  of X don't worry because right here I will have   to tell you guys what the definition of this is  we turn your parentheses this is 1/2 natural log   and we have 1 plus x over 1 minus X so that's the  inverse hyperbolic tangent and we want this add a   equals 0 we may not own this too well but we do  know this very well so attack this right here   right so here we go this right here division is  I can just pretend to two parts so here we have   1 over 2 parentheses that Ln first thing is 1 plus  8 hey don't we have this yes we do right and then   minus Ln 1 minus X like this and so I will put  is plus and negative x yeah okay ah man I wish   I know my power series okay I will try I will try  I will try I promise I will try here we go this is   1/2 and this right here the first thing is let me  just read it out and maybe alternates again so I   already asked a just like to avoid our confusions  earlier so here I was just say this is the sum as   K goes from 0 to infinity it does alternate and  that's negative 1 to the N and ya and then over   n plus 1 X to the n plus 1 power because you  are to integrate the best friends cousin right   so that's what about and yes yeah next this is -  and we have the sum as K goes from 0 to infinity   and we have negative 1 to the nth power over plus  1 but we enter this right here so we get negative   x to the Plus Ones power like this ok but again  they they just like to alternate you know that yeah so let's let's just go ahead and  write down the terms right shall we okay   here we go and guests can come cannot  hear because yeah this is 9:21 p.m. 9/21/2012 a so that's the first term that's  good now when case one you're putting here   you get negative and you get 1 plus 1 which  is 1/2 and then you get x2 yeah next when K   is 2 this is going to be positive and then  there will be 1 over 3 when K is 2 UK X to   the third power and if you do again you get  minus 1 over 4 and you have X to the fourth   power and plus sauce and so on and perhaps let's  do one more just to be safe so plus 1 over 5 X   to the fifth power and then minus also okay  then we are going to - man this is going to   be the opposite of all that so you actually  start with negative verse if K is 0 in here   you see negative x to the first power so it's  negative x and in case 1 well this is negative   now will be negative X so that will give us if  case 1 this will be this to a second power so oh that's that's Britton man let me just break  down the power here this is kind of confusing   so this right here all right this right here  is the same as saying negative you want to the   k plus 1 power right so it's the same as negative  1 to the K times negative 1 to the first and then   we have the X to the K power right so let's  look at this I'm trans I put this tank blue   for you purple for you guys so again negative  1 to the K times negative 1 to the first and   then X to the k plus 1 power so in terms of the  power of the X would be the same but the signs   that knowing part let's take a closer look good  thing is that this and that when you multiply it   becomes just negative 1 to the 2k power which  is positive 1 and the truth is this right yet   not alternating anymore it's just always man  it's just always negative that's kind of bad   but whatever this is always negative so first  of all when K is 0 you get X to the 0 plus 1   who's just X for first power over 1 so that's  what we have and now when X is 1 you put here   so it's x squared and this is still negative so  it's negative x squared over 2 I so I get it so   this is minus 1/2 x squared and then when  X is 2 you get over 3 but this is X star   and this is still negative so it's negative 1  over 3 X cube that's always negative man this   is always negative and minus 1 over 5 X 25  and sauce also my bed for the camera angle yeah okay all right so man this is 1/2 all the  way in the front and now let's see which other   guys that still maintain me okay first of all  this is X minus negative x so that means I will   have 2x right here isn't it next negative that -  the same thing so this and I will cancel that's   good next this minus minus becomes again two of  them so we have plus two over three extra third   power uh-huh pretty good this and that again  they can so and lastly the one that we have on   the board this and that together we get two of  them again so plus 2 over 5 X to the fifth power   and then pass the product like that okay and  of course you can just reduce other tools so   we get X plus 1 third man this is not alternating  but that does this remind you of anything yes you   sure that that's the inverse original tangent but  this one right here is not ultimately that's why   it's the inverse hyperbolic version okay so  finally we'll just say this is the sum as n   goes from I'm using n right here now okay 0 to  infinity no matter dating right so just 1 over   2n plus 1 and there's no factory on either and  just 2 and plus 1 and 2 are numbers and then the   power is the same as that so we have X to the 2n  plus 1 and notice that a is a 0 so that's what we   have right so that's pretty much it now check  this out Oh from here to here well we use the   R is equal to 1 right and likewise from here to  here R is equal to 1 so of course now it's just   algebra so we still know that R is equal to 1  and again you can set out absolute value but   R is equal to 1 and let's see what's I did well  you will be going from negative 1 to 1 now check   this out if I put negative 1 in here this is not  alternating and then negative 1 to the 2 1 it's   just gone it's not alternating so unfortunately  it won't count purge and when we put in one right   here again that will be 1 over 2 and plus 1 you  can just do the comparison test with 1 over N   write the series of 1 over N it does not converge  neither so this open parenthesis open parenthesis   a fast way to do it is that if you put in negative  1 into the original function now I messed up right   here 1 plus negative ones are not 0 in here right  0 itself the aligned so that messed up so you can   hang for that if you're putting 1 you have 1  minus 1 about it again that's messed up for   the function that means you can include that  remember remember remember if the function is   messed up then of course the power suits will be  messed up ok so that's what we were to do it so   you can just do it like this ok so have a look  ladies and gentlemen inverse hyperbolic tangent next one should be pretty fun to do okay next  one Ln of X at two so I'm just kind of jumping   back and forth with the natural log and all that  let's see number 22 I want natural log of X add a   equals 2 so it's a different center now right and  now the different center so here we go this is Ln   of X when a is at you right when a is 2 Center  is 2 that means I want to have a factor X minus   2 well in sense you have X now I have X minus 2  it's okay because I know 2 plus negative 2 is 0   so this right here will be the same and then just  happen like this that's pretty much it right well   in order for us to you know do a quick way I'll  just tell you we will be using our and off and   it's plus signs ready for us too so that's very  nice 1 plus X didn't we just do this one earlier   yes we did this is the sum as n goes from 0 to  infinity and we have this is negative 1 to the   nth power over n plus 1 X to the n plus 1 power  R is 1 and in fact interval notation interval   of convergence is at negative 1 1 and I should  also say Oh F divided by X is less than 1 less   than 1 sorry sir you can just write it down like  this absolute of X is less than 1 R is 1 and the   Interpol convergence a negative 1 doesn't work  by a positive winders yeah I wrote it down wrong   earlier so yeah anyway this is what we are going  to do this is what we are going to do in word of   what you use this friend notice that we must have  a 1 right here but that's a - it's ok just factor   things out yes what we'll try to do isn't I'm  hungry anyway Oh an hour factor or two in the   printing uh inside so I will do this let's say I  have this I have the to me like so and we have one   plus if i factor or two that means what that  means this right here would be X minus 2 over   2 and that's pretty nice close close welcome  to thanks though thanks to natural log Ln of   a product it's the sum of two LNS and this is two  times all this so first we will have Ln of 2 and   then we add Ln of this which is 1 plus and that we  have that which is X minus 2 over 2 I guess this   right here we're just going to deviate right now  Plus well this right here we can use our friend   right here huh so you need to know how to use your  friends 1 maybe 9 your life in math it's okay in   real life man that's kind of not good to say well  in math it's totally ok to know how to use your   friends it's important too all right so here we  have negative 1 to the n plus 1 power over n plus   1 and you just enter these guys in here so we have  X minus 2 over 2 parentheses this and then raised   to the n plus 1 power and I will enter this right  here in the absolute value as well so this radius   of being in red likewise this absolute value  of x minus 2 over 2 should be less than 1 okay   here we have Ln of 2 plus the sum n goes from 0 to  infinity this is negative 1 M plus 1 over M plus   1 well we have 2 to the M plus 1 on the body so  already it's 1 over 2 to the M plus 1 and perhaps   put parentheses to just missing more clear and  help this on the side yeah and now check this   out this is Eleanor - and there's nothing that we  can combine with in that case we'll just leave the   ln of 2 Otto in the front so this right here is  pretty much yet right so we cannot do too much in   this situation and for power series sometimes it's  okay to have like some extra terms in front so   that the rest can be putting the Simo notation if  this one cannot be done then just DV in the front   that's it yeah I know I know yeah anyway right  here absolute value of X minus 2 multiply two on   both sides we have this test and two so that means  our here's two it's ours - then I notice a is to   write a is 2 and R is 2 that means you move to the  left twice and you move to the right twice so you   appeared 0 and you'll be 1/4 so I will be a 0 and  4 but what happened a 0 if you put in 0 right here   you can negative 2 and you unfortunately cannot  do 0 why not because your cannot be put in here   that's a quick way to do it so if you function  can our workforces use cannot work in there okay   you should check for if you put 4 in here you get  party 2 2 to the n plus 1 so this and now cancel   in fact this right here can purchase just like the  one that we have right here so in fact it's pretty   much the same idea open parenthesis closed bracket  it's the same thing just refer back to what with   the earlier if you need to write down all the  steps just go ahead and do so and this is it   right so yeah and of course you can just maybe  write this down one time and maybe yo just put   this down doesn't really matter that much but how  we all put this down like this negative 1 to the   n plus 1 power over I'll put down my 1 times 2 to  the +1 looks better this with neck just like this when that I get a couple more to go not so bad  if you only have to do 26.2 questions hundred   obeah crying imagine 100 power series question  seriously you guys seriously you guys are going   to make me do that seriously because I'm making  me do that now please don't next 123 next 23 23   here we go we have 2x ah this is the polynomial  to express the power and then minus PI X square   plus 1 and we want this to be add a equals  1 in fact you have a non calculus way to do   this I believe you can just do some polynomial  deficient whatsoever but in this case I'll show   you guys a calculator do this again what is that  one that means you should have X minus 1 as your   input right so just keep dying my later on what  you say any better all right anyway I will use   the taylor form lot to help us out and then this  is the derivative here is the formula a 1 over N   factorial a very common mistake that students will  mainly study for care about to defy the factorial   this is not so bad because it's a polynomial if  we differentiate these four times you know it's   over so we don't need too much so let me just  put this down 0 1 2 3 4 I'll show you right   here Ryota so original we have 2 X to power minus  5 x squared plus 1 go ahead differentiate this we   get 6x squared minus 10x differentiate this again  we get 12x minus 10 do you again we get 12x not   just 12 ha ha ha ha and then we get 0 and is he  a potential serious oppa by now put in one here   I want to make sure that I can do this in my head  carefully put him one here this is 2 minus 5 which   is negative 3 plus 1 is negative 2 yes divided  by 0 factorial ok 0 factorial next putting one   in here with your 6 minus 10 which is negative 4  and a deep ID by 1 factorial good then next this   is factorial n factorial and then I'm putting 1  here we get 12 minus 10 which is 2 so we have 2   but divided by 2 factorial and then lastly it's  just 12 because you deal have X anymore divided   by 3 factorial well in fact anything after that  would be just 0 so you don't have to worry about   anything anymore this is not like an infinite  polynomial because originally you start with   a polynomial you will n right so let's see this  right here simplifies to negative 2 so for this   time purple just to make is cooler this is equal  to negative 2 for the coefficient and this right   here is equal to negative 4 2 factorial is 2  and you can see that my hands are really dirty   this right here is 1 3 factorial 6 right because 3  times 2 is 6 and it has 1 of course and this right   here it's going to be 2 and this right here is of  course 0 so now here we go to extra power minus   5 x squared plus 1 let me tell you this one we  cannot really put into the Sigma notation because   should have phone numbers and yeah just ready  out that's all we do so start with this which   is negative 2 and x2 this Rose power nothing X  minus 1 times it was your spot next we do minus   4 times X minus 1 to the first power next we  add one so we just put on plus 1 and X minus 1   to the second power and next one will get plus 2x  minus 1 to the third power like this and the good   thing is that you stop you will stop ok let me  just double check that I didn't make any mistake okay man I don't know why it  doesn't look right on my work okay this is not funny whitey I do that  is that I have a +8 on my answer key for   the this turn so I will double-check as  I said most likely is the one I'm doing   right now it's correct let me just  check on Wolf on Alpha because again   I don't give you guys the one the crow  answers just do this 3 minus 5x plus 1 huh oh yeah yeah I told you yeah so  I will have to fix that okay cool so   that's pretty much it and let me tell you  they are identical polynomial polynomial   so ours of course infinity and has of course  negative infinity to infinity so yeah so this   is what you do with polynomials like this all  right okay just three point two more to go Oh Lucy sigh run your mirrors on that's what I paid  the twenty six point two numbers when the markers   here are just so bad that's why my hands are so  dirty anyway next one this is a famous one this   is number 24 and for this one depends if you  are going over this one not into a class okay   so if you are not doing this you if you're not  doing this in your class than you you don't have   to do it but it's good to know like why not in  so it's math anyway 1 plus X raised to the 8th   power where R is any real number okay and we want  to expand that at a is equal to 0 now let me just   check out a few things ok why is my team not so  much a straight down it's right here where R is   any real ask you the same any real number and  add a equal 0 ok so let me take us a few things   if R is a pasty whole number what maybe it just  it varsity right stupid easy it's just 1 if R   is that say pass the whole numbers I to you can  just multiply getting the polynomial if r3 just   multiply out that's a binomial expansion like  binomial theorem for that this is called the   binomial series if R is I say 1/2 or negative 1/3  or negative 5 whatsoever you can use the following   one so this ratio covers the more cases right hmm  yes you can't use that how you're welcome to use   the thinner from love wait this time it's kind  of weird so I will show you guys how what work   this out for you guys are on the way okay so I  will just write this down as usual and this is   what we have the formula and a is at 0 over N  factorial okay so just like this now check this   out this is kind of weird so just kind of yeah  with me 0 1 2 3 4 here we go the first one is   just the original one which is 1 plus X to the  8th power derivative this I mean differentiate   this you bring the art to the front and a minus  1 yeah so you get R and then 1 plus R and then   1 plus X we are minus ones power like this now  check this out the chandu says you multiply by   DT of thing so which is just 1 so it doesn't  matter and it's a simple of the rest of them   differentiate this again though if you bring  this power to the front and then minus 1 you   get R to R times R minus 1 and then this is 1  plus X 2 or minus 2 right and you can kind of   see the pattern if you just keep going you get  R R minus 1 or minus 2 and for enough space on   minus 3 so that's pretty much the idea for  this series so let me just erase this quick right of course just to detail relief so it's  in you shouldn't be too bad and of course if I   don't do it again then you pretty much walk yet  let me just see if I can feeding this R minus 1   or minus 2 or minus 3 let me let me just operate  so we have three terms of the shopping now right   so that's what we have now check this out when  I put in 0 into X the first thing idea is just   1 and then over 0 factorial so that's good  that's no big deal okay when we're putting   see rolling here I just get R times 1 which  is just R so that's good that's just saw   again nothing crazy and then over 1 factorial  next if I put this right here we get R times   R minus 1 so we get R times R minus 1 because  if you put 0 right here this is just going to   be 1 anyway so we have that and over 2 factorial  and you can guess the next one it's going to be   R times R minus 1 R minus 2 and then divided by 3  factorial and so on so you can guess the next one   is R times R minus 1 R minus 2 R minus 3 over  4 factorial and so on right therefore let's go   ahead and just come with a pattern for this  already when we have n well I'll check this   out earlier when I have Street I start with 1  I stopped it at 2 right so here's the deal if   this is N and let me make this thing purple oh  say if this is let's see how it's already okay   if this is 3 I stop there too so right here I  will get this this will be R and then R minus 1   dot dot dot right and then we have to stop it  add one less there whatever this is this was   three so you stop here one less right so you  step there too so these this is n we have two   minus and minus one like this minus one in the  purple parentheses like this and then multiply   by X to the R - this would be true so yeah so  it's important like this oh 1 plus so I'm sorry   yeah all right so it's pretty much all this code  let me just write it down better yeah so this minus 1 because it's one less right and then you  multiply it by 1 plus X to the R - ends power like   this yeah it's power okay and in the end let  me just write it down like that when you put   in zero in here we'll just get all that which is  I'll put this time in red which is starting with   R and then R minus 1 R minus 2 and so on so on  and so on and the last one we do is going to be   R minus and of course we can distribute this  real quick this is R minus N and then plus 1   so we have R minus n plus 1 now what we did a  relative and don't forget to divide it by the   factorial so divide this by n factorial like this  so in fact this right here this right here is the   formula for the coefficient for that guy right and  the truth is we do have a nicer way to write this   this is how we define already done for you guys  already don't regress so let me just put this   down right here for you guys already this is  the summit and goes from 0 to infinity this is   for the expansion 1 plus X to the a power what  you have is that you write this down which is oh that which is all that right but much better  way to write this down is the following you can   just go ahead and write down our choose and  like so and then you multiply by X to the nth   power yeah and power and like that and again the  center is a zero so just X minus U is just X like   this right and now what's this dope this right  here is precisely that so I would just have to   tell you our choose n is precisely that so is  our R minus 1 R minus 2 that a dot and we have   R minus plus 1 or over and factorial like this  so let me give you some example because this is   ready pizza so here is an example for example if  I have that say if I choose 3 and 2 the way you   can say is 5 to Street what you can do is you  start with 5 you go down next one which is 4   and you go down again which you stop history  right so this tree tells you how many times   you go down so you see like well you you have  to have three numbers sorry it's time you go   you go down once going twice but you should have  three numbers right here so again if you follow   five right that's fine and then 5 minus 1 is 4  and then 5 minus 2 and you stop you at you you   do our minus 2 right here so that's pretty much  it and divided by 3 2 1 on the times 120 and on   the bottom is 6 so no under tab use our 60 on the  bottom stick so it's 10 so that's an example let   me give you guys another one so suppose you  have 1/2 choose 3 I didn't prepare this I'm   doing this in my head so we'll see if we have  1/2 choose 3 this is what we do you start add   1/2 you start from 1/2 next you do 1 half minus  1 next you do 1 half minus 2 so again you have   to have three numbers and then you / three  two one name these three factorial and then   we can just do the math in your head we'll just  just do the math this is one-half times this is   man this is negative one half yeah then this  is negative three over two yeah and all over   six okay this is eight on the pattern is 48 and  we have a three on the top and this is positive   it's also get is 1 over 16 I believe this is the  answer so let me just check on one help out real   quick on this so the way you do it is just  say one-half choose 3 hey I got it right so that's pretty much the idea behind the  coefficient right here and again this   is the proper way that you should expend  it what's the radius of convergence I can   tell you this one but I don't know if I  should prove it off I should so I will do   this real quick so you can just write it  down like this okay you can just read it   on it I think totally that yet well what's  the radius of convergence you have to take   the limit as n goes to infinity and here  we go this is going to be the following take an apple for you this is our C n so  we will just have our choose n times the   reciprocal of that so you just have to put it  as one over and we will have cm plus 1 so the   end it becomes n plus 1 so we have our M plus  1 like this all right so let's see if I have   enough space to do all this or not here is the  limit as n goes to infinity what's our choose   n do we remember this right here is R times  R minus 1 dot dot times R minus n plus 1 ok   / and factorial okay now for the reciprocal  version you know you have to divide it by n   factorial so just put the sorry n plus 1 factorial  so just put the n plus 1 factorial on the top and   then you do this pretty much so you have R times  R minus 1 dot dot dot times R - what well this   right here again is r minus n minus 1 1 less than  why ever this is now we do 1 less than whatever   this is so it would be just our minus n like  that yeah right so let me see that is correct   of course okay now let's see how we can do for  this first of all this right here is n plus 1   factorial name the n plus 1 times n times n minus  1 so also that's n factorial this and that can be   cancelled it that's very nice then let's see this  is our minus n and again this was and this is our   minus and minus 1 in the parentheses right so the  truth is there is a factor near here R minus and   minus 1 namely a term P for this right because  this is study mode it has more term so we do   have this factor right here as well you have a  factor right in front this is ends term this is   the n minus 1 term the good thing is that all  here will be cancelled with everything here so   what we have left this is the limit as n goes to  infinity al-adha is just nice T equal to n plus   1 yeah and on the bottom is just this term which  is R minus n notice that we do have the absolute   value because technically this is from the ratio  test take the limit inside out you get negative   1 in the absolute value which is just positive  so let me tell you this right here aha is equal   to one very nice right and usually for this right  here to find out the convergence at the endpoint   of the interval right because we go from negative  1 to 1 this right here it depends so I'll just do   this right here have you seen this notation do  you guys know what this notation is you don't   write because yeah I don't need it because I just  make this up this right here okay this right here   the pants are at depends on are depends on R so  I watch how you you let me just see you include   the this right here so you can just see it yeah  yeah so I will tell you if R is in between of   negative 1 and 0 then the interval notation  interval of convergence is negative 1 and 1 including this but not including that yeah and then if the R is bigger than or equal  to 0 then the interval of convergence is both   negative 1 and positive 1 and otherwise  I will just be negative 1 1 9 2 the end   points so for example best friend right here  it's technically when R is equal to negative   1 and that it's actually right here and if you  have the best friend if you differentiate that   again you will be right here as well so that's  that yeah and I do notice this is 1 plus and   I'm the best one - but it's a similar idea you  can just do the substitution but man this is the binomial series man ok so have a  look oh yeah of course of course   of course but this I'm going to erase  this right huh and this takes me how   many hours already this 3 hours man  I can almost go to figures already all right so we have a couple more to go or as  I'll just be a deal with me well I'm doing this   for you cause no no no not Tony's it yet this  is 25 so here we go square root of 4 plus X add   a is equal to 0 well of course we'll be using  our the formula we derived earlier so you can   see that this right here it's the same as a  parenthesis 4 plus X to the one-half power   but in water for us to use for we did earlier  and perhaps them already down to remind you   guys we will be using the binomial 1 plus X to  the art power that's the sum as n goes from 0   to infinity and I'm just going to write here and  so our choose n times X to the nth power like so   and then the radius of convergence is 1 which is  up to 5 X is less than 1 it's another way to say   that of course and of course here is the X in  red yes the X in red here is the X in red right   now our is 1/2 but we need that one so you know  the teal we can factor things out so here we go right here factor out the 4 so we have the 4  right here and this is going to be 1 plus this   is bit going to be ah x over 4 like this and  then we have 2 1/2 which is the square root of   course and of course this right here is the same  as saying what would be 1/2 power you just read   yes what would be 1/2 power times 1 plus X over 4  to the 1/2 power like that this right here is the   same as 2 very nice this right here it's going  to be the sum as n goes from 0 to infinity and   we'll just be using that right here so we will  have our is 1/2 and the an and we'll just enter   this into the X which is x over 4 raised to the  nth power like this good isn't it ok so here we   have to sum those series yes and goes from 0 to  infinity just ready down like that 1/2 and and   let me write this down as 1 over 4 it's the  same as 2 to the 2nd power and then the nth   power already yes 2 to the 2 and power because  I have the 2 right here I want to do this so we   can match their powers the base and this right  here's X to the nth power like this now this is   on the top is 2 to the first power technically  I should do 1 minus that right but I can do this   this is the sum as n goes from 0 to infinity 1/2  choose n how to already has a rider 1/2 chosen oh   yeah put out doesn't really matter hmm yeah let  me just put on this further mom is a key harder   this part first by anyway times 1 over well I can  just do the bottom - the top and they stay on the   bottom that's what you can do as well so it's 2 to  D 2 minus 1 so you understand the bottom this is   what you do and then in the end you have X to the  nth power like that very very very nice now R is   equal to well let's do that haven't done that yet  so who the update your absolute value of X minus 4   this has to be destined one more to tie for both  sides up to 5 XS destined for R is for radius of   compared I mean the center says Europe so I it's  equal to negative 4 to 4 well what do we say   about intervals earlier when R is positive like  well 0 as I said grayton include 0 u in coupons this is when R is greater than equal to zero  include both include both imports and again   if R is a [ __ ] I wrote but like if is a in  between of negative 1 and 0 then just include   one in point I'll type here for you guys ready  just go to read this code look at that okay so   just yeah okay so that's number 25 two more  to go I'm going to still use that so yeah   this one's going to take me another like ten  minutes or so I don't know I can do the easy   work and do the hard way I don't know so number  26 you know this is going to be the hardest one in purse thanks at X equal to at a equal to 0  a the center so yeah yeah well how can we get   a inverse sign yes we can use what integration so  reminder so this is just a little reminder right   because if you are taking care to maybe haven't  seen the integral for this for a while but if you   integrate 1 over square root of 1 minus X square  this right here gives you impressed sign yeah   so a strategy is come up with a power series for  this first and then integrate that so here we go   so that's the need we need a funnel so we you  want okay we want one over square root of 1   minus X square first so that's one one first and  here we go square root on the bottom we can bring   that up and that's a negative 1/2 power so this  is actually the same as 1 I need to see the plus   I have to see a plus it's ok just put a plus and  that's a negative x squared and then raised to   the negative 1/2 power like this very very nice  isn't it it can be hard it can be easy depends on   how you look at it so we have 1 over square root  of 1 minus x squared this is going to be throwing   negative x squared into here and there so we have  the series and goes from 0 to infinity or in our   case is negative 1/2 so negative 1/2 choose N and  then just enter that negative x squared raised to   the nth power like this include the absolute  value of negative x squared that's the one   then we integrate both sides so integrate this  integrated neato DX 0 DX the left hand side give   us exactly what we want that's the inverse sides  is equal to C plus don't forget the C plus here   integrate this are may I forgot to simplify this  a little bit okay let me just do this right here   real quick right this right here is negative 1 to  the nth power yeah and then X to the 2n power so   of course simplify this we have been simplifying  this for like 26 25 times already yeah all right   so look at a new power work which is 2n plus 1  and divided by that so that's pretty much just   divided by 2n plus 1 so we get the sum and goes  from 0 to infinity let me write down negative 1   over 2 choose n times this right here is what  we need for the negative 1 to the N over this   2n plus 1 and that's a new power study two plus  one power like this and that's very good yeah   and find out what C is that's x equals zero when X  is equal to zero well we get what we get M inverse   sine of zero that's equal to C plus when we're  putting zero into here everybody will be zero   okay because this is zero to the first power one  nine zero right when n is zero get one so it's   zero to the first power and everybody else has the  X and everybody will be zero so in other words C's   in co2 0 so C is equal to zero like that okay  so that's pretty much it and yeah this is the   little multiplication and you may be thinking  this isn't so bad right why and I think this is   the long version let me just write it down nicely  for you guys the Sun and goes fast you wrote in   that heat this is negative one over two and and  this is negative one and power over two and plus   one it's just a lot of writes not so hot seriously  okay okay what I said this is like the hardest one   it's because some people may not take this as the  correct answer I don't know why they just want to   break it down I can break it down for you guys as  well so let's go ahead and do the breakdown here   let's observe how this notation is going to look  like check this out negative one half choose and   it's going to be negative one over two that's  your first thing okay and then times negative   one over two minus one and then negative 1 over 2  minus 2 and then dot dot dot and then negative 1   over 2 - let me just - and then plus 1 after that  right that's the definition shows you are you /   factorial okay so let me see if I have enough  space to work this out I hope I hope I do now   check this out this is going to be negative one  half for the first term and this right here if   you add of subtract the fractions come by the  fractions you get negative three over two okay   and for the next one you get negative five over  two and da da da ya ya now if you look at this if   you just get a common denominator you of course  multiply this by 2 and know it's like by 2 and   this there was no parentheses so I should not I  should do this not to do this what apply this by   2 and 2 and then 2 & 2 so ya power series power  series man everybody has a 2 on the top here okay on the top here negative 1 plus 2 is 1 so we  have 1 but this is a minus 2 so it's minus 2   and plus 1 so here are your factors in all that  stuff okay over N factorial now we're going to   break this down work noticed on the table is  negative 1 negative 3 negative 5 and so on so   on so on how many terms do we have in fact on the  top you also have n terms just like the example of   the say 5 choose 3 when you have this you to  5 and then you write down a total of 3 terms   right here if you have a 10 choose 6 you will  have 10 and then that you have six numbers so   the truth is on the table you also have in terms  so yes negative 1 everybody has a negative 1 this   right here I can have a negative ones will look  at this as negative parenthesis to add and minus   1 right so you have a negative 1 right here as  well therefore I will have a negative 1 to the   nth power for that and then we will have 1 3 5  up to two and minus one so that's that this and   then everybody has the two on the bottom so we  have the two to the nth power on the bottom so   this right here is two to the nth power and  then we have this which is M factorial right   so oh this right here is to show you this right  here is the same as that okay so that's what we   have and if you want you can continue I will  continue because it's making me mad for this   question I was trying to figure out but like oh  yeah I didn't know I've tried to read this down   for you guys nicely let me see I'm dropping that  space by the way R is one so just just just just   do that okay just shake the answer for that I'm  not going to do that we're going to conquer this   guy so here we go this is sine inverse of X this  is the Sun goes from zero to infinity I believe   if you write the answer down like this nobody  can be mad at you seriously seriously okay but   if you want to simplify it more here is how you  can do it this is the sump and go smash you're   rotting in that heat now here we go I will put  all this down right here and that's one of the   answers to half and this is actually answer from  the textbook I believe so I will just put this   down right here so we will have this which  is negative 1 to the nth power and then all   that so it's 1 3 5 dot a dot 2 a minus 1 okay and  then on the bottom here we have 2 to the n times   n factorial okay so that's what this one notice  that we still have this part which is negative   1 to the nth power over 2m plus 1 and then X to  the 2n plus 1 power ok so that's what we have now negative one to the N times negative one to  the N is just negative 1 to the 2 and that   means everybody would be positive so you end up  with this this is the sum as n goes from 0 to   infinity and you just get 1 times 3 times 5 times  all that stuff I'm gonna tell you guys one of the   studies what this is right here is what we call  the top of factorial this is an odd number so   what you can do is you can write this down as 2  n minus 1 factorial factorial that this is Kadapa   factorial quick example it has to be a number  inside in that case you get a say 7 factorial   you just go down by 2 every single time all these  numbers are so you for example 7 factorial 7 5 3   1 like this and working out and if you have that  say 9 factorial is of course a nice 7 yeah I lost   creativity right now also a table factor tapo tapo  factory okay and with this notation you have to be   careful though you can prove will you usually  you cannot you can use the definition negative   1 tamo factorial it's actually 1 okay I will do  a video for you guys if you like negative on top   of factory off by definition is equal to 1 where  any reasoning that you want isn't equal to 1 so   when you put in 0 here you end up with negative  on top of factorial it's 1 right so again this   right here just merge you into two and minus one  type of factorial and then this guy on the bottom   we have over and I believe the textbook answer is  yeah it's just this so we have 2 to the N and 2 n   plus 1 and then n factorial so it's like this  and now we'll have the missing black 2 plus 1   and we still have that X to the 2n plus 1 so this  is totally okay again right so again look at this   look at that and you kill this now if you want  to simplify more which I have another one for   you guys if you want to simplify more the truth  is this is 2n 2 to the N times unpacked for real   you can read it down like the sum and goes from  0 to infinity on the table use 2 minus 1 double   factorial and there on the bottom this and  that is the even type of factorial so it's 2   and double factorial and because of the factorial  so I will actually write down this first 2m plus   1 and then this and that together becomes a 2  an double factorial and then you have the X to   the 2n plus ones power man and then for that one  factorial what you do is well it's just there's a   six double factorial is just six times four times  2n stuff you don't go to 0 and work that out 0   type of factorial is equal to 1 I have another  video you guys want to check that I have that   for you so this is by definition and the other  one just how you do it so why don't we just stop   right here I really don't know R is equal to 1 and  let's see what's my radius of convergence oh yeah well I will let you guys tell me what's the plot  the interval of convergence right here right so at   is it enough of this you guys leave a comment down  below okay so this is number 26 so have a look so   that's it way that's it or that's it so depends  on how do you want to present your answer up to   you right now I will just write down the last  one for you guys here now you guys know as you   finish a marathon the worst part is not a 26 mile  it's the last point to my own okay so here is the   last point to 26.2 and here 1/2 X to the 0 point  2 power add a equals 26 ok this one's actually go   pretty quick here we go X to the 0 point 2 well I  need to have a whole factor X minus 26 so let's do   chaos inside X minus 26 well this is X so what we  have to do is just add a 26 back and that race was   real point to like so now we have the 26 right  just factor out so factor other 26 and we have   1 plus this is X minus 26 over 26 and then well  you guys know just using your head all right this   is going to be raised to the 30.2 power and this  right here is going to be a risk to a 30.2 power   as well all right so this is going to be let's  see this is 26 and this is the slow point to I   will leave it right here 26 to the zero point to  power this right here use the power series sum   as n goes from 0 to infinity here is zero point 2  choose an and then the employees X minus 26 over   26 grace to the end power right and of course do  the after the fire all that but let me tell you   our is the equal to 26 okay now we just have to  simplify this a little bit we were to something   your head okay as I've done enough of all the  details for you guys already leave it seriously   talk to see see - you see leave it so you're going  to choose that we will have 26 to the nth power on   the bottom but then here we have the 0.2 power  on the top so we just had minus 0.2 and this is   1 and then we have X minus 26 choose n power  man let me tell you we are done RS 26 and the   interval of convergence start at 26 left right  so you have 0 + 52 this is past deep are right   0.2 so you include both and points oh man this is  how you finish the marathon seriously to finish   everything non-stop in one go that's how you do  a marathon right so how long did this take me   three and a half hour man my hands are all dark  because of the marker and also the dirt and all   that stuff but I still would like to put on my  my arms are so tired me entering something first haha I have time marathon meadow this is the  Long Beach marathon now I just ran up a like   a stew weeks ago yeah Long Beach marathon  hey so I'm going to put this down this is   actually pretty I don't I didn't want to  touch my matter with my left my right hand   because is dirty you guys can see I messed  up but oh well let me show you this is how   to 26.2 power series implant I do now will be my  protocol thumbnail right so anyway hopefully this   helps you and leave a comment down below if you  guys have any questions and best of luck to you   guys in your car to class and if you guys are  new to my channel hopefully you can subscribe   and I don't know how many you thumbs up with  this power I feel really good just I haven't   have finish marathon I feel pretty good so yeah  just for the record anyway as always that's it

Channel: blackpenredpen

Views: 90,825

Keywords: Power series marathon, Power Series of cos(x) at a=pi/4, Power Series of cosh(x), Power Series of tanh^-1(x), Power Series of tan^-1(x), Power Series of ln(x) at a=2, Power Series of sqrt(4+x), Power Series of sin^-1(x), power series examples, power series study guide, power series of sin(x), power series of e^x, power series of cos(x), power series representation of functions, interval of convergence and radius of convergence, blackpenredpen, binomial series

Id: LKhvdkUdLtE

Length: 216min 9sec (12969 seconds)

Published: Tue Nov 05 2019