My problem is to compute series $\sum_{n=1} ^\infty \frac {1} {n^4} $ and $\sum_{n=1} ^\infty \frac {1} {(2n+1) ^4} $ using Fourier Series for $ f(x) = \lvert x \rvert^3, \lvert x \rvert \leqslant 2 $.
I have found the Fourier Series for $ f(x) $ to be as follows:
$ f(x)=2+\sum_{n=1}^\infty \frac {48[(n^2 \pi^2 - 2)\cos(n\pi) + 2]}{n^4\pi^4}\cos(\frac {n\pi}{2}x) $
In the picture below, $ f(x) $ and it's Fourier Series are plotted together.
Any ideas of how to compute these two series?
P.S:
$ \sum_{n=1} ^\infty \frac {1}{(2n+1)^4} $ was obtained by dividing the mentioned Fourier Series into odd and even parts of n, considering $x=2$ and the result is:
$ \sum_{n=1} ^\infty \frac {1}{(2n+1)^4}=\frac {\pi^4}{96}-1 $
But still don't know how to compute $ \sum_{n=1} ^\infty \frac {1}{(n)^4} $.