Infinite Series - Numberphile

Key Takeaways

so i'd like to talk about three infinite series so sums of infinitely many numbers so here's the first one one plus a half plus a fourth plus an eighth plus blah blah blah what's that and the answer is it's two this is the problem of achilles and the tortoise achilles is two stadia or whatever the unit of length is in ancient greece from the tortoise well after a while achilles gains on the tortoise one unit and so the remaining distance is only one then after a while achilles gains half of the distance to the tortoise and now the distance to the tortoise is a half after a little while achilles gains another one quarter of the distance to the tortoise and so on and so on and so on the story according to zeno's paradoxes is that achilles never catches the tortoise well of course achilles never catches the tortoise at any of these times but if you just wait a little longer achilles will have caught the tortoise the total length that achilles gains on the tortoise is two and that's why this is true now let's look at one plus a half plus a third plus a fourth plus a fifth plus a sixth plus a seventh plus an eighth and so on and this never ends how big is that what is that what i mean is if i stop after many many many terms so i can so i'm adding some huge finite list of numbers how big is that and what happens when i add more and more and more of them and the answer is this is infinity why well let me group the terms here's the first one here are the next two here are the next four the next eight would end with one over fifteen and this goes on forever i don't wanna mark up my desk professor you said infinity i'm always taught that's not a number like well so what exactly does this mean you give me any number at all 50 trillion and i can stop this series and say okay i will only go this far i'm adding only a finite number of numbers together but that sum will be more than 50 trillion that's true not only for 50 trillion but for any number you name no matter how big and that's what it means that the sum of the series is infinity so it's blowing up it's like exactly it's getting away from us exactly we say that the series diverges so let's look at the first term well it's one thank you uh how about these next two terms well you know what this one is bigger than a half each one of these terms is bigger than a quarter a half is bigger than a quarter a third is bigger than a quarter and there are two terms so this is bigger than two quarters which is again a half how about these four terms they're all bigger than one-eighth and there are four of them so if i add these guys it's bigger than four-eighths which is a half if i look at the next eight terms they're all bigger than 1 16 and there are eight of them so the sum is bigger than 8 16 which is a half if i stop far enough this sum will be more than a half plus a half plus a half plus a half plus a half and i can go on as many times as i like accumulating one half until i get to more than 50 trillion so sooner or later the sum of these numbers will be more than 50 trillion the catch is that i have to go enormously far out in order to do it but let's not worry about that now okay never mind just how enormous that's let's leave it there for this sum which has a name it's called the harmonic series i'd like to explain how to use the harmonic series to stack dominoes well actually i don't have dominoes in my office but i have lots of issues of the annals of mathematics oh yes i have lots and lots and lots so we're going to use copies of the annals of mathematics as dominoes let's just take two of them first if i move this one here it's going to collapse but if i move it so that its center of gravity is let's see if that works i'm being a little conservative the center of gravity of this guy is sitting over that guy and so it does not collapse now let's take these two and put them very carefully on top of the third one taking care that the center of gravity of this whole guy is above this issue of the annals of mathematics and therefore this looks precarious okay there it does not collapse and now perhaps i could put these three on top of a fourth and if i succeed in doing that i will not tempt fate i will just declare that uh that we have succeeded whoop so let's see how are we doing it hasn't collapsed looks a little flaky it hasn't collapsed how far can we make this guy stick out suppose that i have all those copies of the annals of mathematics or perhaps all the copies of the annals of mathematics ever printed from the beginning of time to the end of time wonder how many that will be anyway we have all of them and we stack them up and we demand that the pile cannot collapse well this is merely paper so sooner or later if there's too much of a load it will be crushed but never mind these are imaginary rigid dominoes can we ever make the top guy so far out that it is not resting on the bottom guy at all that this edge is not lying at all over this book but has come out farther further out in the bottom book exactly and the answer is oh yes in fact you can make this this pile come out in that direction as far as you like because if you look at it if you have n copies of the annals of mathematics and you do an absolutely perfect job in an ideal world the distances that they are slid out from one another are in the proportions one to a half to a third to a fourth to a fifth to a sixth and so on to one over n if there are n copies of the annals of math or maybe n plus one copies of the annals of math and so the fact that the harmonic series diverges tells you that you can eventually make this pile come out as far as you like because one plus a half plus a third plus a fourth plus and so on if you stop way way far out after a gazillion for a large enough value of a gazillion will be more than 50 billion and therefore this distance out there will be more than 50 billion times the height of the annals of mathematics all right let's put the annals of mathematics aside now let's do the same thing to the series one plus one half squared plus one third squared plus one fourth squared plus let me keep going fifth squared okay blah blah now let's try to do the same thing we're going to take the first term all right so that's 1 thank you let's look at the next two terms now this is 1 over half squared this is smaller than 1 over a half squared so we've got two terms and they're each smaller than one-half times one-half so that's smaller than two times a half times a half that's smaller than a half let's look at the next four terms here they are they're all one over four squared or smaller so they're smaller than one-fourth squared plus one-fourth squared plus one-fourth squared plus one-fourth squared there are four terms there and so these guys together are no bigger than 4 times 1 4 squared that's 4 times 1 4 times 1 4 and that's 1 4. so this whole thing is less than 1 4. if i looked at the next eight terms and played the same game i would find that that's less than one-eighth and so on so now look if i go out to a gazillion terms i'm guaranteed that what i've got is less than one plus a half plus a fourth plus an eighth plus a sixteenth plus something or other i stop at one over some power of two hey wait a minute remember uh first result this series never i mean the sum of these guys is never more than two the sum of all these numbers is two and the sum of the first gazillion numbers will be slightly less than 2. so therefore i'm guaranteed that no matter how many terms of this guy i take it will be less than 2 so it doesn't run off to infinity the way the harmonic series does and in fact you could wonder what is the value of the sum of all of these terms and that was a famous unsolved problem of the 18th century and the answer is pi squared over 6. pi creeps in where you would least expect it that you're going to get the medal i was exactly here this place and i was having an interview then the phone rings hello oh and the voice was saying hello this is laszlo lovas from budapest