Infinite Geometric Series & Intro to Limits in Calculus - Part 1 - [18]

well welcome back the title of this lesson is called the infinite geometric series this is part one of several I'm really excited to teach this lesson because of a couple of reasons one it has a kind of a counterintuitive lesson it doesn't make sense at first but I will absolutely make have it make 100% sense by the end here but more importantly it's because it dovetails very very nicely into calculus so here we're kind of at an algebra precalculus level but if we are going into calculus soon and so I get to introduce some calculus topics here I do not believe in babying you I do not want to introduce too much calculus too early however when there's a good opportunity then I'm not gonna pass that opportunity up and that's what we have here so you're going to learn a little bit of some sort of very elementary calculus concepts the biggest one here that you need to kind of wrap your brain around is when we talk about the infinite geometric series we're talking about a series of terms that are added together that a geometric series right but an infinite number of them not ten of them not 50 of them not a thousand of them not a million of them not twenty quadrillion of them literally an infinity of them so well first time you think about that it should kind of blow your mind how can you add an infinity of things together I mean that means there's an infinite number of things that means I can continue adding them till the end of the universe and I'm still not done adding them so how is it possible that I can add up these guys at all and furthermore in this lesson we're gonna find out that when you add up terms of a geometric series under a certain condition I'll tell you in a minute then not only are you adding an infinite number of terms but if you were allowed to do that you're actually gonna get a number a single number how can that be if you add an infinite number of things down well shouldn't the answer be infinity I mean if I'm adding forever shouldn't be answer be infinity in other words I always never get to the answer well actually that's not true and that's super important for you to conceptualize because calculus is completely based on this idea the whole branch of calculus that you learn the second half of calculus one is all about adding up an infinite number of things it's crucial to our modern ideas of engineering and math and science that we use out all the time to calculate things so we're kind of inching our way to those ideas in this lesson all right so we need to back the truck up a little bit talk about the infinite geometric series what I'm gonna do is write down the sum the actual finite sum of an infinite geometric series is gonna write it down and then I'm gonna back up the truck and take you from the beginning so that you understand exactly where this equation comes from and then we're going to apply it to lots of problems the good news is it's very easy to understand and the problems are very very simple but I do need you to watch this lesson entirely so that you will be with me all the way to the end all right so what I want to do first like I said is I want to give you the punchline first okay it turns out that I can add up and find the sum of an infinite an infinite set of terms in terms of a geometric series so when you see the word infinite in front of geometric series it means the terms never ever end now in order for a geometric series to be to be when you add them up to actually get a number out of it a single number okay there's a very important constraint remember in this geometric series we have what we call the common ratio so we have to say that if the common ratio the absolute value of it is less than one okay then the following is true the sum of notice I didn't say S sub n I mean the sum of all infinite all infinite number of those terms is equal to the first term divided by one minus the common ratio right so let me back up a little bit and talk about this when I say if the absolute value of R is less than 1 I need to talk about what this means what this means is the following it means that the common ratio has to be greater than negative one okay and the common ratio has to be less than one okay it's very confusing when you see it written like this but almost every book is gonna they're not gonna write it like this everybody can understand this R has to be greater than negative one and R has to be less than one that makes sense to me bigger than negative one right in less than one in fact if I put this on a number line what I'm basically saying here is if this is the common ratio and this is 0 and this is 1 and this is negative 1 what I'm saying is that common ratio has to be here it has to be a fraction a negative fraction less a little bit bigger than negative one and a fraction a little bit less than one in other words the common ratio for this to work has to be like 1/2 or 3/4 or point nine nine nine nine nine nine nine nine just a little bit less than one or it can be negative 1/2 or negative 3/4 or negative point nine just just like this if you make the common ratio anywhere in here where it's a fraction smaller than one either positive or negative then you can find a sum of the series and when I say the sum of the series I mean a sum of all infinity terms that never ever end that's what's so crazy about it but it's absolutely true and I'm gonna show you an example why that's the case and that sum is going to end up becoming the first term divided by one minus this common ratio now one thing I need to point out before we get too much farther is that when you have a constraint like the common ratio has to be bigger than one in less than one all it's telling you is the common ratio has to be a fraction less than one but either positive or negative remember common ratio is telling me how to find the next term in the next term in the next term if the common ratio is bigger than one outside in this region if the common ratio is five 5 for instance that means I'm multiplying by five and the terms are getting bigger and bigger and bigger if I add up an infinity of terms that are getting always bigger then of course I'm gonna get infinity I'm not gonna get an actual sum it's gonna be no sum it's gonna become infinity so the only way you can get a sum an actual number is if the common ratio is a fraction that means the terms are getting smaller and smaller and smaller that's the only way it works if the common ratio is one-half for instance that means I'm multiplying by one half to get each next term so multiplied by 1/2 multiplied by 1/2 and so on so all the terms are going down and that's when you can say the geometric series sum is equal to a number that's why the common ratio has to be in that window the negative if it's a negative fraction it's the terms are still going to get smaller they're just gonna alternate signs but they're still gonna get smaller if the ratio is over here the terms will be positive and they'll get smaller if the common ratio is anywhere outside of this region like negative 2 or positive 2 the terms will get either bigger or they'll get bigger but alternating bigger the negative R means it's gonna multiply by negative so it's gonna get bigger it'll get alternating bigger in order go out of control so you won't have a sum when you see it written like this what it's saying is if you were to put like let's say negative 0.5 in here take the absolute value that's less than 1 if you put a positive 0.5 in here take the absolute value it's still less than the 1 so writing it like this it's just a shorthand way of saying this okay so enough of that I don't want to talk about that anymore that is the punchline that's what the sum of the geometric series is what I want to do now is show you why and not just with a simple little proof I want to give you a real example so what I want to do is let's consider a very easy to understand geometric series so consider the following geometric series we've looked at this before 1/2 plus 1/4 plus 1/8 plus 1/16 plus dot dot dot when I say dot dot dot that means there's an infinity of terms after this term would be 1 over 32 1/32 then it would be 1 over 64 then it'll be 1 over 128 there'd be one over 256 it would go on and on and on forever and ever and ever but notice that these terms are getting smaller why are they getting smaller because what is this common ratio how do I go from this term to this one I multiply by 1/2 1/2 times 1/2 is 1/4 how do I go from here multiply by 1/2 1/4 times 1/2 is 1/8 how do I go from here to here again I multiply by 1/2 this times this gives me this so the common ratio here is 1/2 right and that means that the common ratio is less than 1 the absolute value of it so if you put a 1/2 in here take the absolute value it's less than 1 check that means that we already know ahead of time that this series is going to have a sum and if I wanted to calculate the sum which I'm not going to do yet but we will do it all I would have to do is put the first term in here 1 minus the the the common ratio that we just figure it out and we can calculate this sum like that I don't want to do it yet I want to show you why it works ok but we know that the common ratio criteria is correct in other words it's 1/2 the common ratio is right in here and that means these times are getting smaller and that means this is going to have a sum and by the way when we say a series has a sum an infinite series has a sum we say that the series converges when you say the series converges it means it has a sum even though there's an infinite number of terms when you say the series diverges that means there's no sum it just goes off to infinity and lots of series also diverge as well so let's take a look at this in more detail that's what I want to do next let's look at the partial sums remember that we talked about that partial sums what is the S sub 1 that means the sum of the first term well if you look at the first term that's only one half so there's nothing to really add okay but if you were to convert this fraction to a decimal it's exactly equal to 0.5 okay now let's look at the second partial sum the sum of the first two terms and that means it's going to be 1/2 plus 1/4 because that is the first two terms how do I add these ok just to make sure everybody's on the same page I could say well I've got 1/2 I'm adding to it 1/4 I need a common denominator so I'm multiplied by 4 over 4 and so what am I gonna have actually it's not going before before sorry about that working a little bit too fast it's gonna be I can multiply by 2 over 2 so this will be 2/4 plus 1/4 when you get here 3/4 okay and as you have 3/4 as the answer the exact value is 0.75 because 3/4 you all know is 0.75 let's take a look at the third partial sum what will it be 1/2 plus 1/4 plus the third here is 1/8 okay how do you do all of this well you need a common denominator and you'll have to find a common denominator of with an eight and an 8 here and I'll just do it one more time to make sure we're on the same path here we have 1/2 1/4 1/8 so to get a common denominator I multiply here by 4 over 4 multiplied by 2 over 2 so what am I get I get here four eighths okay here I will get two late and here I have the 1/8 so when I go back over here I'll put an equal sign right below so 4 plus 2 is 6 plus 1 is 7 8 that's the common denominator here and whenever you convert this to a decimal 7/8 you're going to get 0.875 all right bear with me we're actually almost done let's take a look at the fourth partial sum 1/2 plus 1/4 plus 1/8 plus what is that fourth term 1/16 all right we'll get the common denominators I'm not gonna do it all but you have a common denominator of 16 that you could get multiplied all these guys add them all up what are you gonna get you'll actually get 15 16 so I encourage you to do that on a separate sheet of paper and when you crank through this and get a decimal what do you get 0.9375 I think you can see what's happening here but just to bring the point home let's skip down so there's a fifth partial sum and a sixth and a seventh let's add the first 10 terms up the tenth partial sum so what will you have it'll be 1/2 plus 1/4 plus 1/8 plus 1/16 plus dot dot dot and then well let's put a plus sign here 1 over 1024 because the last term when you multiply by 1/2 ten times is going to be 1 over 102 4 if you get a common denominator of all these and add them up you would get 10 23 over 10 24 you see this is really really close to 1 in fact what do you get here it's gonna be 0.99 902 and this is I'm gonna put approximate because this is rounded what have we done here we started out and said here's an infinite geometric series as an infinite number of terms I'm telling you ahead of time this has a it converges in other words you can add an infinite number of terms and it have actually equal a number right if we only add the first term up well in other words only consider the first term we get 0.5 if we add the first two terms this is the answer if we add the first three terms point 875 we add the first four terms 0.9375 first 10 terms 0.99 902 what do you think is gonna happen if we add the first thousand terms cuz this is an infinite series we we got to the first 10 terms and already got a number really close to 1 what's gonna happen if we get to the first 15 20 terms what about a thousand terms what if we take a million terms what if we consider 20 quadrillion billion kajillion that's not even a number but if we looked at all those terms what's gonna happen is you're gonna get point nine nine nine nine nine nine you're gonna get really really really really close to a number this series converges you consider more and more and more terms trying to get as close to infinity as you can and all that happens is the sum gets closer and closer and closer to a finite number that finite number is actually going to be equal to one so to put that in words okay this is where the calculus part comes in what we say is as the number of terms that we consider approaches infinity this arrow means you can never get to infinity you didn't just consider more and more and more terms like I was telling you verbally but if when we consider more and more more terms then the partial sum when you put infinity down here is going to approach the number one will it ever quite get to the number one well no because we can't ever calculate an infinity of terms but conceptually the closer we get it always approaches closer and closer and closer to a finite number than number one so theoretically if you could add an infinity of terms it would exactly equal the number 1 that means the series converges if it were to go up to infinity when we add up these numbers then it would diverge so we say that this thing converges and so in the language of calculus right so in the language of calculus whoops I can spell calculus correct this is what we say we say that this thing has what we call a limit when we let the limit as the number of terms go to infinity of the partial sum of the number of terms in other words we put in as 1 in this 5n is 3 in its you know 35 it is 100 and as a million and so on this limit equals 1 if we let in actually approach infinity it gets closer and closer and closer so we say that when n becomes infinity and in fact it can never be infinity but as it approaches infinity this limit becomes one okay so will kind of put this guy right here and we say this series we say it converges in your algebra book if you're looking at an algebra book it'll probably won't say that it'll probably say the series has a sum or there's a finite sum to the series but in the language of calculus you say the series converges in that case and you say that the limit of the sum of the partial sums as n goes to infinity is is equal to 1 it's equal to a finite number so as an example of when you can add up an infinity of things but get a finite number but notice the only reason it works is because these terms are getting smaller if these terms are getting bigger like if this series were 2 4 6 8 10 then as I add an infinity of them together of course it's going to blow up it's gonna diverge that's what will happen but that's not what's happening when you have a geometric series where the TUC the terms are getting smaller and smaller and smaller then we say the series has a finite answer but why exactly does it approach one I mean obviously we can we can write it down we can see that it approaches when we calculate it but fundamentally let's go a different angle and let's try to figure out why it approaches one so I'm gonna put a big Y here that's what we're gonna investigate next I want to talk to you about Y so I want you to recall the nth partial sum we talked about geometric series already we said we can add up a finite number of terms and we said that that sum the finite sum was called S sub n and it was T the first term times 1 minus R to the power of n over 1 minus R this is the the equation for the nth partial sum that we have derived before if I want to figure out what the first three terms of the series is that's okay I put the common ratio in put n is equal to 3 I know the first term crank through it and I'll get the sum of the first three terms in fact I could use that equation to calculate the second partial sum the third the fourth 2/10 if you put these numbers in here you will exactly get back the numbers that we got by hand here that's exactly what you would get all right but let's go through it for the case when we're trying to go to an infinity of terms with this common ratio that we have so in this case what we have is the first term in our series if you go back to our series the first term was 1/2 so we'll put T 1 is 1/2 1 minus the common ratio what was our common ratio the common ratio was 1/2 we kept cutting the terms in half every time so the common ratio was 1/2 power of n is still right here on the bottom we have 1 minus R we just said the common ratio was 1/2 so what's going to happen here S sub n the sum of the nth term is going to be 1/2 let's go in here and say 1 minus 1/2 again to the nth power on the bottom 1 minus 1/2 what do you get 1/2 so what's going to happen is you have a coefficient of one half on the top of one half on the bottom so what actually happens here is the nth partial sum the only thing you have left is this 1 minus 1/2 to the power of n what does this equation tell you this is telling me if I have a geometric series that starts with the with the first term of 1/2 and that has a common ratio of cutting every term in half each time in other words it's for our specific series that we just looked at on the board then the sum that you get when you consider the first n terms is equal to this so this means that when when n gets very very big right what's going to happen when n gets big this is 1/2 to the N power if it's 1/2 to the 10th what's going to happen it's 1/2 times 1/2 times 1/2 times 1/2 and you do that 10 times so when n gets bigger and bigger and bigger this thing gets smaller so as n gets bigger what happens is 1/2 to the N gets smaller why because it's a fraction raised to an exponent so it's 1/2 times 1/2 times 1/2 as in and gets bigger bigger bigger then this thing gets smaller smaller smaller so as n approaches infinity this sum approaches 1 because this thing is getting smaller and smaller and smaller and so the only thing left is one so I just wanted to show you like it's it's one thing to just write the terms out and to say oh yeah it looks like they're getting close to one that's cool but it's another thing to take a different angle and say this is the equation that we already used for the in number that sum the sum of the first n terms we put in the information for our specific series with our specific common ratio in our specific first term and what you get out of that is something that looks like this and it's very clear that as n gets bigger all that's happening is the sum is getting closer to one that is why the sum is getting closer to one over here because the equation we already learned about to calculate the sums approaches one as n gets bigger and bigger and bigger now what I want to do is I want to check my math here and make sure that I didn't forget anything so I guess the one last thing I can say so we can say that 1/2 plus 1/4 plus 1/8 plus 1/16 plus dot dot top we can say that that's actually equal to the number 1 but implicit in this is that whenever you add the dot dots you're adding an infinite number of things beyond it only when you consider all infinity of terms do those the sum actually converts to 1 if you stop at n is equal to 3 million terms then it's gonna be real close to 1 but it's not going to be 1 if you consider all infinity of terms which kind of blows your mind then you can actually get to 1 ok and now you can see why it's so critical in the beginning here what did I tell you I said the sum of the infinite geometric series is going to have this let's we're gonna come back to this this is gonna be the sum but in order for the sum to exist at all the common ratio has to be less than one I mean this means less than positive 1 or bigger than negative 1 it has to be a fraction somewhere in this region that's what it's saying now you can see actually why that's the case the common ratio went in here in our case it was 1/2 so when you go through the math the common ratio pops out here and so what's going on is if the common ratio is a fraction then when n gets bigger this term goes down down down down down as the terms get bigger but if that common ratio were not a fraction let's say the common ratio were two let's say then let's say let's put it like a two in here instead of one half to me and it was like two to the end when the common ratio is too big that means that this term never gets smaller it just gets bigger and then the whole thing blows up to infinity and the series does not converge that is why the common ratio has to be a fraction to force the terms to go down close to zero then in that case you can add an infinite number of terms together and get a number if the common ratio is outside of that of that range then the terms don't go down and then the series does not convert and just to kind of illustrate that a little bit more directly let's consider another series let's consider another series consider consider 1 plus 10 plus 100 plus 1,000 plus dot dot dot this is a geometric series how do you know well because to go from this term to this term I have to multiply by 10 to go from this term to this term multiplied by 10 those terms of this term multiply by 10 it's a geometric series so you might say well I'm gonna add them up and figure out what what happens here but notice that the common ratio here is equal to 10 and that's way outside of what I said had to be the ratio has to be between negative 1 and positive 1 this thing's way outside of that and because of that as n approaches infinity is that look at what happens if I start adding all these terms then the sum approaches infinity as well it does not converge so does not converge in other words there's no sum because as you include more and more terms the sum blows up instead of getting smaller so geometric series can add up to a finite number but only when the common ratio is between negative 1 and positive 1 that's the bottom line when the common ratio is between negative 1 and positive 1 the terms of the series are going down and because of that there's a finite sum and that finite sum is equal to this now the last part of what I want to do is show you why the sum actually equals this so to figure that out to prove it take a look at that let's take a look at a proof of this remember the impartial some we talked about that before it was t 1 1 minus R to the power of n over 1 minus R we covered this a couple lessons ago you've learned it it's the sum of the first in terms of a geometric series okay but I want you to consider the case consider if n goes to infinity all right we have to consider the first infinity terms right and if that happens if absolute value are less than 1 that means the common ratio is between plus or minus 1 right then what's going to happen here in this sum is R to the power of n is in that case going to go 0 why because if the common ratio is in the range that I'm saying it has to be it has to be a fraction like put a 1/2 in here then 1/2 to the N is I let n go to infinity would be 1/2 times 1/2 times 1/2 times 1/2 so if you do an infinite number of times this term right here goes away so what will happen is it'll be something like this SN T 1 1 minus R to the power of n over 1 minus R if all this is true in fact I can kind of say well let's consider an infinite number of terms and then we'll put infinite number of terms here but what's going to happen if R is a if R is a really a fraction like we're saying and you raise it to that power of infinity this term just goes to 0 in the case when the common ratio is small like that if the common ratio is bigger than 1 then I raise it to an infinity power then it's going to get huge it's gonna go to infinity but in the case when it's a fraction like this it doesn't it doesn't happen so what am I have left the sum of everything is gonna be the first term times 1 which means tyonne's on the top and 1 minus R is on the bottom this is exactly what I told you the sum is equal to so the reason why the sum of an infinite geometric series of terms is equal to this is because when you look at the sum that we learned before for a finite number of terms and consider the case when a common ratio is a fraction then if I let infinity of terms in there this whole entire thing just drops away it goes to 0 and all I'm left with is this is kind of the proof of that you just examine the case when the common ratio is a fraction and let the number of terms go to infinity boom that term drops away all you're left with is this so what I would like to do now is let me sum I have room over there yeah let's do it over here let's go back to our example that we started the lesson with and we started the lesson and we said consider the following series we said what about 1/2 plus 1/4 plus 1/8 plus 1/16 plus dot dot it's an infinite series and we said we did the terms we know what it comes out to be it comes out to one right we already said this but let's apply this we're saying the sum is going to be equal to in the case when the common ratio is a fraction between plus or minus 1 we know it is because we've already done that the sum is gonna equal to the first term which is 1/2 divided by 1 minus the common ratio which the common ratio again we've already said is 1/2 so what do you get 1/2 over 1/2 so what does this mean it means the sum of all this stuff is 1/2 divided by 1/2 which is 1 this is the same sum that we calculated in the very beginning we took these terms we literally did it by a brute force method of taking more and more and more terms adding them up and we figured out that it approached 1 and we did a bunch of talking and kind of looking at how it all works and have a look at that equation and how everything works out and it all works out to 1 but then we say here's a theorem you can use it for any geometric series if you know the first term and you know that the common ratio is between plus or minus 1 then this is the sum all you mean is the first term in the common ratio when you have an infinite number of terms you add them up that's what you get we apply it to the same exact sum that we started with the begin with and we get exactly the same answer so I'm kind of trying to draw in a big bow for you to show you that the equation works the theorem works and it matches with everything we've talked about already so that's all I have for this lesson we're gonna do a lot more problems in the next couple of lessons mostly what I want you to understand out of this is conceptually what's happening it is completely possible to add up an infinite number of things but yet get a finite answer as a result then something that you just have to wrestle with and you have to come to accept because all of calculus is based on it you will get to that much later when we study calculus but the second half of calculus one is all about essentially you're looking at sums of things and I'm not going to get into it but that's what we call an integral in calculus it's an infinite sum of tiny little things so you have to get comfortable with the idea of adding things together so this concept here of this infinite series of geometric series giving you an actual finite answer goes directly into some of the concepts of calculus so make sure you can understand all of these I would work through this if you can or at least play it a couple of times to make sure you get it then follow to the next lesson we're gonna start cranking through problems the actual equation is very very simple you just have to verify that that common ratio is a fraction like we discussed and you're good to go to use the equation here so follow me onto the next lesson we'll get more practice right now

Channel: Math and Science

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Keywords: geometric series, infinite geometric sequences, infinite geometric series sum, geometric sequence, arithmetic sequence, arithmetic series, calculus, algebra, precalculus, pre-calculus, infinite sum, geometric progression, arithmetic sequence formula, geometric sequence formula, geometric series formula, summation notation, infinite series, sequences and series, ilecture online, infinite geometric series, geometric series test, geometric series sum, series convergence, convergence

Id: a61SUx6EtM8

Length: 29min 47sec (1787 seconds)

Published: Tue Jan 19 2021