# Fourier series and reimann zeta

1. Dec 22, 2011

### hectorzer

1. The problem statement, all variables and given/known data

Find the Fourier series expansion for f(x)=$x^{3}$, a periodic function on -$\pi$<x<$\pi$
Use this to compute $\zeta$(6)=$\sum\frac{1}{n^{6}}$

2. Relevant equations

Parsevals Theorom,
Real Fourier series

3. The attempt at a solution

I got the Fourier series to be $\sum\frac{2(-1)^{n}(6-n^{2}\pi^{2})}{n^{3}}$sin(nx)

Using Parsevals theorom I got that $\frac{\pi^{6}}{7}$=$\sum\frac{4(6-(n\pi)^{2})^{2}}{n^{6}}$

The answer is supposed to be $\frac{\pi^{6}}{945}$ I think, I can't see where I went wrong :S
Thanks in advance :)

Last edited: Dec 22, 2011
2. Dec 22, 2011

### hectorzer

Never mind, I got it.
Just had to expand the top and use the values for the riemann zeta functions of zeta=2 and 4 and it works out :)