Boundary Values & Fourier Series · Challenge problem

Find the Fourier series of $f(x) = |\sin x|$ on $[-\pi, \pi]$ and apply Parseval's identity $\dfrac1\pi\displaystyle\int_{-\pi}^{\pi}f^2\,dx = \dfrac{a_0^2}{2} + \sum_{n\ge1}(a_n^2 + b_n^2)$ to find $\displaystyle\sum_{m=1}^{\infty}\dfrac{1}{(4m^2 - 1)^2}$.