Evaluate infinite sum using Parseval's theorem (Fourier series)

[thaer_dude]

Homework Statement

Show that: $\sum_{n=1}^{\infty}\frac{1}{n^4} = \frac{π^4}{90}$
Hint: Use Parseval's theorem

Homework Equations

Parseval's theorem:

$\frac{1}{\pi}\int_{-\pi}^{\pi} |f(x)|^2dx = \frac{a_0^2}{2}+\sum_{n=1}^{\infty}(a_n^2+b_n^2)$

The Attempt at a Solution

I've been trying to solve this for ages and I just can't figure out what to do. I know you're supposed to use Parseval's theorem. All I've managed to do was plug in $\frac{1}{n^4}$ into the summation part of the Parseval's equation and I substituted the formula for a₀ but I couldn't get very far.

Any help would be really appreciated.

 

[micromass]

Try the function $f(x)=x^2$ for $x\in [-\pi,\pi]$.

 

[thaer_dude]

It works, ty