Boundary Values & Fourier Series · Hard problem

The Fourier series of $x^2$ on $[-\pi,\pi]$ is $\dfrac{\pi^2}{3} + \sum_{n\ge1}\dfrac{4(-1)^n}{n^2}\cos nx$. 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 $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^4}$.