Fourier series and calculating a sum

[toofle]

Homework Statement

Studying for an exam, this is a question from an earlier exam:

Calculate: sum k from 1 to infinity, 1/k^2

I have from previous question:
f(t)=pi+sum n from 1 to infinity: 2*(-1)^n/n*sin(nt)

f(t)=pi-t, -pi<=t<pi

So i need to rearrange this to calc the sum.

Homework Equations

The Attempt at a Solution

calc f(t) in a good point, I choose pi/2, use the left and right limit to get pi/2

then try top rearrange the series.


Especially I want to know mhow to do Sin(n*pi/2). it makes every second term disappear in the series but I don't know how to do this right. I tried replacing with n=2k+1 but it doesn't work.

 

[susskind_leon]

The idea behind this kind of approach is to expolit Parseval's identity.
If f(x) can be expressed in terms of a Fourier series with coefficients $a_n$, then you have
$$\int_{-\pi}^\pi |f(x)|^2 dx = \sum_{n=-\infty}^\infty |a_n|^2$$
Can you find your original series somewhere in what you have obtained through the previous question?