Euler's summmation of 1/n^2

Date: 03/15/2000 at 23:51:55
From: Brad Bennett
Subject: Euler's summmation of 1/n^2

How do you go about proving that pi^2/6 = the summation of 1/n^2 from 
1 to infinity? What mathematical method could I use to solve the 
problem?

Date: 03/16/2000 at 16:38:46
From: Doctor Schwa
Subject: Re: Euler's summmation of 1/n^2

This is a hard problem!  A good discussion can be found in George
Polya's excellent book, _Mathematics and Plausible Reasoning_.

A short answer:

Consider the function sin (pi*x). (I know you're asking "what does
sin (pi x) have to do with the sum of 1/n^2?" but bear with me).

On the one hand, it's a function that's zero for every integer k.
So, as an infinite polynomial product, it must be true that
sin (pi x) = a x (1-x) (1+x) (1-x/2) (1+x/2) (1-x/3) (1+x/3) ... 
so that the right-hand side has zeros in all the right places. 
By difference of squares, we get
sin (pi x) = a x (1-x^2) (1-x^2/4) (1-x^2/9) ...

Now the left side can also be expanded in a power series,
sin (pi x) = (pi x) - (pi x)^3 / 3! + ...

Using that, we first discover that a = pi to make the x coefficients
match up, and then discover that the sum of 1/n^2 is pi^2/6 to make
the x^3 coefficients match up.

Feel free to write back with questions about all the steps I skipped 
in this derivation. It's a really neat problem!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/