Expand $f(x)=x, 0<x<\pi$, as a half-range sine series.

By integrating the series show that a Fourier cosine series for $x^{2}, 0<x<\pi$, 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

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