Euler's solution to the Basel problem

1. May 6, 2010

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Can someone find a good explanation of how Euler did it? I can't seem to find anything article or whatnot that carefully explains what he did to solve the problem. From what I can gather, he seems to follow a method of writing out sin(x) as an infinite series(taylor polynomial) and divides by x and from there I get lost in the mishmash. Why does he even choose to use sin(x) and not cos(x) or tan(x) for that matter?

2. May 6, 2010

Count Iblis

The function

sin(pi x)/(pi x)

is zero at the nonzero integers, so it should be equal to:

sin(pi x)/(pi x) = product over n of (1-x^2/n^2)

The right hand side is a product that converges and it is zero when x isa nonzero integer. The normalization is correct, because for x = 1 it is 1 while the limit for x to 1 of the left hand side is also 1. So, the above indentity looks correct (but you can't rigorously prove that, Euler only conjectured the identity).

The coefficent of x^2 of the left hand side is:

-pi^2/6

And from the right hand side it is minus the sum of 1/n^2 from n = 1 to infinity. To get an x^2 term, you need to take it from one factor of the infinite product, say the nth, and then you need to take the 1 from all other factors. You then get -1/n^2, and all n from 1 to infinity contribute.

Now, you can just as well do this using cos(pi x). The zeroes are at x = (n+1/2), so you would conjecture that:

cos(pi x) = Product over n of [1-x^2/(n+1/2)^2]

Normalization is correct as can be seen from putting x = 1 on bith sides. Extracting the coefficient of x^2 gives:

-pi^2/2 = -Sum over n from n= 0 to infinity of 1/(n+1/2)^2

You can rewrite this as:

sum over n 1/(2n+1)^2 = pi^2/8

You then use the following trick. If we put:

Zeta(2) = sum from n = 1 to infinity of 1/n^2

Then clearly the sum of the inverse squares of the even numbers only is:

sum from n = 1 to infinity of 1/(2n)^2 = 1/4 Zeta(2)

So, the sum over only the inverse squares if the odd numbers must be
3/4 Zeta(2). So we have:

pi^2/8 = 3/4 Zeta(2) -------->

Zeta(2) = pi^2/6.