Fourier series of x cubed minus pi squared x
= Fourier series of x cubed minus pi squared x
{c}
On $(-\pi,\pi)$,
$$
x^3-\pi^2x
=12\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}\sin(nx).
$$
Parseval's identity and direct integration of the square give
$$
\sum_{n=1}^{\infty}\frac1{n^6}=\frac{\pi^6}{945}.
$$