Boundary Values & Fourier Series · Superchallenge problem

Find the Fourier sine series of $f(x) = x^4 - 2\pi x^3 + \pi^3x$ on $[0, \pi]$, and use it to evaluate $\displaystyle\sum_{k=0}^{\infty}\dfrac{(-1)^k}{(2k+1)^5} = 1 - \dfrac{1}{3^5} + \dfrac{1}{5^5} - \cdots$.