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ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12
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CEFR Level
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I'm gonna give you an astounding result
Astounding?
An astounding result
So I'm just gonna write down a little sum
I'm just gonna see what answer it gives.
1 + 2 + 3 + 4 + duh duh duh duh duh
And I include all the natural numbers, so all the way up to infinity
So what do you reckon the answer to this is, Brady?
-Well I would say it would go to tend towards infinity.
Yeah, that makes sense, doesn't it.
The answer to this sum is -- remarkably -- minus a twelfth.
It's amazing! I mean, I first saw this result when I start learning a bit of String Theory
And what's even more bizarre is that this result is used in many areas of physics
This is a very well known string theory textbook by Joe Polchinski. As you can see,
sort of quite early on, page 22, we have this statement here which is that the sum of
all this is --- basically saying the sum of all the integers ---
all natural numbers all the way up to infinity, is, minus a twelfth.
Alright so we're going to prove now,
Without getting our knickers in a twist with Riemann zeta functions, we gonna prove
in quite simple way, why the sum of all the natural numbers is indeed minus a twelfth.
So, to do that we're gonna do this in a number of steps.
We're gonna to look at a few different sums.
The first sum that I want to look at, I'm gonna call it S1.
And it's 1 -1 +1 -1 +1 -1 and so on.
That's the first sum I wanna look at in principle.
The second sum I'm gonna look at is 1-2+3-4 and so on
Carry on in that process all the way up.
And the third one is of course going to be
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