I have a piece-wise function as follows: $$ f(x) = \left\{ \begin{array}{ll} 1 & \quad 0 < x \leq {\pi\over2} \\ 0 & \quad {\pi\over 2}< x < \pi \end{array} \right. $$ I am supposed to sketch the even and odd extension of $f$ and compute the Fourier Series for each.
I got $$f_{even}(x)={a_0\over 2}+\sum_{n=1}^{\infty}a_n\cos(nx)$$ $$a_n={2\over \pi}\int_{0}^{\pi}\cos(nx)dx$$ $$n=1,2,...$$ $$a_0={1\over 2\pi}\int_{-\pi}^{\pi}f(x)dx$$
and $$f_{odd}(x)=\sum_{n=1}^{\infty}b_n\sin(nx)$$ $$b_n={2\over \pi}\int_{0}^{\pi}\sin(nx)dx$$ $$n=1,2,...$$
I want to determine whether or not the Fourier Series converges uniformly to $f$ for each. I think it does, since we know that if $f$ is periodic, continuous, and has a piece-wise continuous derivative then the Fourier series associated to $f$ converges uniformly. I don't know if there is a way to see this without doing a calculation or if it's necessary to work it out.