Let \(f(\theta)\) be the \(2 \pi\)-periodic function determined by the formula

\[f(\theta)=|\sin \theta|, \quad \text { for }-\pi \leq \theta \leq \pi\]

Show that the Fourier series for \(f\) is given by

\[\frac{2}{\pi}-\frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\cos 2 n \theta}{4 n^{2}-1}\]

From this, show that

\[\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{4 n^{2}-1}=\frac{\pi-2}{4}\]