Consider the step function $f(x) = \begin{cases} -1 & -\pi < x < 0 \\ 1 & 0 \leq x < \pi \end{cases}$
Calculate the coefficients $a_n$ of the sin series $f(x) = \sum_{n=1}^\infty a_n\sin nx$.
What is the pointwise limit of this series, for each $x \in (-\pi,\pi)$?
Does this series converge uniformly? Prove that it does or does not.
I've managed to do the first part, $$a_n = \frac{2(1-(-1)^{n})}{\pi n}.$$ But I am getting confused with all the information available on the web regarding pointwise and uniform convergence of Fourier sine series. I believe I showed that for part 3 we have uniform convergence based on some estimates of the series but I would like to know the general result for these sorts of objects.
In general, if you have good references on calculus like Fourier Analysis for beginners please share them. Thank you!