I heard 1/2+1/3+1/4+... goes to infinity but I didn't think it would go past 3. Reddit r/learnmath

[Music] why does this sequence go to Infinity I  heard the sequence 1/2 + 1/3 + 1/4 + 1/ 15th D   dot goes to Infinity why is that I really didn't  think it would go past like St or something well   let's go for this firstly though this right  here is not sequence a sequence is just a   list of numbers with commas this right here is a  series because we have a sum right 1 12 plus 13   so on so so secondly this right here very famous  series called the harmonic series and the full   version of it is 1 + 1 12 plus 13 and so on but  we don't have the one here anyway though this   right here still goes to infinity and I would to  address your concern that this right here does go   above three because it goes to Infinity so let  me show you here's a quick note if we just add   1 over 2 plus 1 over3 plus dot dot dot up to 1  over 31 of course this is just a finite amount   of fractions we can add them out regularly or  you can just use summation on W Alpha let's use   n start with two and ends at 31 and then we have 1  over n know work this out whichever way you'll get   approximately 3.27 which is of course above three  yeah and now let me show you the classic proof   that why this right here diverges to Infinity  diverges means that it does not converge to a   finite number all right check this out in fact  this right here give you a little hint this is   almost a power of two but anyway though here's  a proof of it I'm going to start by writing down   1/ 12+ 1/3 + 1/4 and then plus 1 15 and then I'm  going to end up with the next power of two which   is 1 8 and then 1 over9 and then the next x^ of  two is 1 over 16 and then 1 over 17 and then so   on so on so on yeah so of course this is the  same as that I just write down like more terms   now H let's make an observation 1/2 is 1/2 I would  like to just keep that I'm just going to keep that   but I'm going to look at this and that together  have a look we know 1/3 about 0.33 this is about 0.25 this is bigger than that so I can replace  the 1 over3 as 1 over4 and I will just make this   thing be greater than that so instead of 1 12  + 1/3 I will be looking at this as 1 12 plus   1 over4 and I'm going to keep the one over4 as  how it is right and then next here is the trick   1 over 5 well this right here will be bigger than  1 over 8 right 0.2 this is 0 something about that   bigger I will replace this by 1 over8 and we have  a greater than already so we can legitimately say   that this right here just replace it and this  inequality is true and in fact in the middle we   will have a total of four of the one over8 I'm  just going to keep it as the same fashion yeah   and then right here same thing what are we going  to replace this by yeah you guessed it 1 over 16   so plus 1 over 16 and then da da da up to this 1  over 16 and then lastly this right here is what   1 over 32 and then so on so on so on all right  now here is the beauty of this I'm going to put   on equal I'm going to put on equal sign because  we know 1 over 4 + 1 over 4 is just 1/2 so this   two together is just 1/2 but we have a first one2  right here yeah next we add this 1/2 yeah and then   here remember we have 1 over five so that's the  first 1/8 and next we have 1 over 6 so that will   give us another 1 over 8 and then 1 over 7 that  will give us another 1 over8 and then one more   1 over 8 all together we have 4 over 8 right just  add them up regularly and of course that's 1/2 so   we add one2 to it as well yeah and perhaps I will  just purposely put down the spacing so that we can   see better here's the 1/2 plus this right here is  1/2 plus this right here we also get 1/2 and then   you guessed it this two here we also end up with  1/2 and then of course course we can continue we   get 1/2 have a look once we get to here let's see  this is one two and the next one is three because   up to one over 31 32 right yeah in inquality work  check this out how many one halfes do we have   this right here keeps going forever right we will  have infinitely many 1/ half so of course when we   add infinitely many 1 halfes we get infinity let  just make a not here uh because uh we have we have   infinitely many halves we have infinitely many  half yeah so as you can see this hormon series   without one goes to infinity and this right here  is the classic proof of it and if you would like   to see this from a geometry point of view I also  show you that as well so check this all right here   we go so of course I'm looking at 1 over n which  is like 1/ X so I'm going to just take a look at   the graph of 1/x like this this is just y = 1 /x  and I'm going to produce these numbers from this   picture check this out I'm going to have 1 2 3  4 five six and so on so on so on all right and   to give 1/2 this is how we can do it just go to  the x value two and then go up to the Y and there   you have it this right here is2 and because the  bottom right here is just withd one so the area   here is just 1/2 yeah and then next one over three  you just go to x value three go up and then make   a rectangle so you can see that in fact I will  just label this because the area is 1 12 because   it's 1 * 1/2 and then this is 1 * 1/3 so this is  1/3 and then 1 over 5 1 over four sorry 1 over   4 and then 1 over 5 and so on so on so on so have  a look we know that 1/ 2 + 1 over 3+ 1 over 4 and   then plus 1 over 5 and so on so on so on this  right here will be greater than well these are   the areas of the great angles and we can clearly  see that we have this part as sticking out right   well it's going to be greater than if we just  have the area under the curve so if you look at   just the blue part right here the area under the  curve in blue I'm just going to put this down as   what integral yes starting at two to Infinity for  the function 1 /x DX and how do we do this well   we get get natural log and I'm will just work  this out we get natural log of x and then we go   from two to Infinity when we put Infinity into  natural log we have Ln of infinity right taking   a limit all right this is just a Shand notation  for that and then pluging to you get this this is   finite but this right here is what Infinity this  right here is about uh 0.693 infinity minus this   of of course is infinity so this right here is  infinity as you can see this will be adding up   to be bigger than infinity therefore this has to  be Infinity as well happy face yeah just like that

Channel: bprp calculus basics

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Length: 8min 47sec (527 seconds)

Published: Thu Feb 08 2024