Let $f(\theta)$ be a $2 \pi$-periodic function with Fourier expansion

$$f(\theta)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)$$

Find the Fourier coefficients $a_{n}$ and $b_{n}$ for

$$f(\theta)=\left\{\begin{aligned} 1, & 0<\theta<\pi \\ -1, & \pi<\theta<2 \pi \end{aligned}\right.$$

Hence, or otherwise, find the Fourier coefficients $A_{n}$ and $B_{n}$ for the $2 \pi$-periodic function $F$ defined by

$$F(\theta)=\left\{\begin{array}{cc} \theta, & 0<\theta<\pi \\ 2 \pi-\theta, & \pi<\theta<2 \pi \end{array}\right.$$

Use your answers to evaluate

$$\sum_{r=0}^{\infty} \frac{(-1)^{r}}{2 r+1} \quad \text { and } \quad \sum_{r=0}^{\infty} \frac{1}{(2 r+1)^{2}}$$