$$f(x)=\begin{cases}\tfrac{1}{2a},& 0<|x|
Note that $a<\pi$.
$$a_0=\frac{2}{a}\int_0^a \frac{1}{2a}dx=\frac{1}{a}$$
$$a_n=\frac{2}{a}\int_0^a \frac{1}{2a}\cos\left(\frac{n\pi x}{a}\right)dx=\frac{\sin(n\pi)}{an\pi}$$
There are no $b_n$ coefficients because the function is strictly even.
Hence the Fourier series representation is:
$$f(x)=\frac{1}{2a}+\sum_{n=1}^\infty \frac{\sin(n\pi)}{an\pi}\cos\left(\frac{n\pi x}{a}\right)$$
But the infinite series on the right collapses to zero, so something must be wrong here. Can someone help me out?