Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function of period $2 \pi$ such that $f(x)=x^{3}$ for $-\pi \leqslant x<\pi$.

The Fourier series for $f$ has the form $\sum_{1}^{\infty} b_{n} \sin n x$. Find
$$
\sum_{n=1}^{\infty} b_{n}^{2}.
$$


Using Parseval's Identity 

$$
\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty}\left(a_{n}^{2}+b_{n}^{2}\right)=\frac{1}{\pi} \int_{-\pi}^{\pi} f^{2}(x) d x
$$

and the fact that all $a_{n}=0$,

$$
\sum_{n=1}^{\infty} b^{2}=\frac{1}{\pi} \int_{-\pi}^{\pi} x^{6} d x=\frac{2}{7} \pi^{6} .
$$

