# proof of the convergence of a series of Fourier coefficients

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.

• What kind of theorems do you know that guarantee convergence of Fourier series? – mrf Oct 4 '15 at 22:18