Expand $f(x)=x$ as a Fourier series on $-\pi<x<\pi$.

By integrating the series show that $x^{2}$ on $-\pi<x<\pi$ can be written as

$$x^{2}=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cos n x$$

where $a_{n}, n=1,2, \ldots$, should be determined and

$$a_{0}=8 \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}} .$$

By evaluating $a_{0}$ another way show that

$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$$