Fourier series of a function defined by parts

*I'm having trouble with the following question. Any help is appreciated.*

Calculate the Fourier series for

$\displaystyle f(x) = \left\{\begin{array}{rl}

x^2, & 0 \le x \le \pi \\

-x^2, & -\pi \le x \le 0,

\end{array}\right.$

and $\displaystyle f (x+2\pi) = f(x), \forall x \in \mathbb{R}$. Given that $\displaystyle \sum_{n=1}^{\infty} \frac{1 - (-1)^n}{n^4} = \frac{n^4}{48},$ find the value of $\displaystyle \sum_{n=1}^{\infty} \frac{1 - (-1)^n}{n^6} =

\sum_{n=1}^{\infty} \frac1{(2n+1)^6}$