Proof of the value of an infinite sum

[reb659]

Homework Statement

On an exam we were given that the \Sigma 1/n^2 was (pi^2)/6. We were asked to prove that the value of \Sigma ((-1)^(n+1))/(n^2)) = (pi^2)/12.

I'm sorry about the lack of latex; this is the first time I've ever *tried* to use it.

Homework Equations

Basic properties of series.
\Sigma 1/n^2 = (pi^2)/6

3. The Attempt at a Solution
I rewrote the series as:
\Sigma 1/(2n+1)^2 - \Sigma1/(2n)^2
From the given information I know that \Sigma 1/(2n)^2 = (pi^2)/24.
The problem is that fact that I don't know how to find \Sigma 1/(2n+1)^2.
I used the integral test to approximate it, and I got 5/4, which is extremely close to the value it should be, which is (pi^2)/8, but I don't think this counts as "proving" it.

Can anyone think of an alternative way to go about proving it, or finding the exact value of \Sigma 1/(2n+1)^2?

 

[Dick]

The sum of 1/(2n)^2 + the sum of 1/(2n+1)^2 = the sum of 1/n^2. Isn't it?

 

[reb659]

...I can't believe I overlooked that. It was sitting in front of me the whole time :(.

Thanks a ton.