[16 pts] Let $f$ be the piecewise defined function given by $f(x) = \begin{cases} \frac{2x}{a} & , 0 < x < \frac{a}{2} \\ \frac{3a - 2x}{2a} & , \frac{a}{2} < x < a \end{cases}$, where \\
constant.
(a) Sketch the graph of $f$ on $0 < x < a$. (Do this by hand.)
(b) Sketch the odd periodic extension of $f$ for three periods. (Do this by hand.)
(c) Find the Fourier sine series of $f$. Simplify your answer as much as possible.
(d) What does the series converge to at $x = 0$, $x = \frac{a}{2}$, $x = a$, and $x = \frac{3a}{2}$?
(e) By evaluating the series in (c) at an appropriate value of $x$, determine what the series $\sum_{n=1}^{\infty} \left( \frac{6}{(2n - 1)^2 \pi^2} + \frac{1}{(2n - 1)\pi} \right) \sin\left( \frac{(2n - 1)\pi}{2} \right)$ converges to.
