Boundary Values & Fourier Series · Challenge problem

Let $f(x) = 0$ on $(-\pi, 0)$ and $f(x) = x$ on $(0, \pi)$. Find its Fourier series on $[-\pi, \pi]$, then use Parseval's identity $\dfrac1\pi\displaystyle\int_{-\pi}^{\pi}f^2\,dx = \dfrac{a_0^2}{2} + \sum_{n\ge1}(a_n^2 + b_n^2)$ together with $\sum\dfrac{1}{n^2} = \dfrac{\pi^2}{6}$ to find $\displaystyle\sum_{k=1}^{\infty}\dfrac{1}{(2k-1)^4} = 1 + \dfrac1{81} + \dfrac1{625} + \cdots$.