So I'd like to talk about three
Infinite series so sums of infinitely many numbers
So here's the first one 1 plus 1/2 plus 1/4 plus 1/8 plus
Blah blah blah. What's that? And the answer is it's 2 this is the problem of Achilles and the tortoise
Achilles is to
Stadio 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 1
then after a while Achilles gains
Half of the distance to the tortoise and now the distance to the tortoise is 1/2
After a little while Achilles gains another 1/4 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 2 and that's why this is true
Now, let's look at 1
plus 1/2 plus 1/3
Plus 1/4 plus 1/5 plus 1/6 plus 1/7 plus 1/8 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 for the next eight would end with 1 over 15, and this goes on forever
I don't want to mark up my desk
Well, so what exactly does this mean you give me any number at all
50 trillion and I can stop this series and say ok. I will only go this far
I'm adding only a finite number of numbers together, but that sum will be more than D 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
Exactly. We say that the series diverges. So let's look at the first term. Well, it's 1. Thank you
How about these next two terms? Well, you know what? This one is bigger than 1/2
Each one of these terms is bigger than 1/4
1/2 is bigger than 1/4 1/3 is bigger than 1/4 and there are two terms. So this is bigger than
2/4 which is again 1/2. How about these four terms?
They're all bigger than 1/8 and there are 4 of them. So if I add these guys it's bigger than
4 eighths which is 1/2
If I look at the next eight terms, they're all bigger than 1/16 and they're eight of them
so the sum is bigger than
8/16 which is 1/2 if I stop far enough
this sum will be more than 1/2 plus 1/2 plus 1/2 plus 1/2 plus 1/2 and
I can go on as many times as I like
Accumulating what have 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 enormous ly 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
looks
precarious
Ok 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 that we have succeeded
Whoop. So, let's see. How are we doing?
It hasn't collapsed
Mm-hmm 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 nevermind these are
imaginary
rigid Domino's
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
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
but if you have any 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
1/2 1/2 to 1/3 to 1/4 to 1/5 to 1/6 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 1 plus 1/2 plus 1/3 plus 1/4 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 1 plus 1/2
squared plus
1/3 squared plus
1/4 squared plus let me keep going
Squared ok, blah blah. Now, let's try to do the same thing. We're gonna 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 1/2 squared
This is smaller than 1 over 1/2 squared
So we've got two terms and they're each smaller than 1/2 times 1/2. So that's smaller than 2
times 1/2 times 1/2
That's smaller than 1/2. Let's look at the next four terms. Here. They are
They're all 1 over 4 squared or smaller
So they are smaller than 1/4 squared plus 1/4 squared plus 1/4 squared
Plus 1/4 squared there are 4 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 1/8 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 1 plus 1/2 plus 1/4 plus 1/8 plus 1/16
plus
Something-or-other. I stopped at 1 over some power of 2. Hey, wait a minute remember first result
This series never be the sum of these guys is never more than 2 the sum of all these numbers is 2 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
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
I was exactly here this place and I was having an interview than the phone rings
Hello and the voice was saying hello, this is Las Navas from Budapest