proof of the convergence of a series of Fourier coefficients

Question

Let $a \in (0,1/2]$ and define $ f:\mathbb T\rightarrow \mathbb R $ by $$ f(x) = \begin{cases} 1, &\text{if $x$ is between $-a$ and $a$} \\ 0, &\text{otherwise} \end{cases} $$

I figured out all the Fourier coefficients of $f$, but the question asks me to show the sum of all these Fourier coefficients converges as an infinite series to $f(0)$. Could someone help me with this problem? Thank you so much.

Answer 1

You can use Dini's test in this case, since the function is pretty simple. MathWorld