Convergence of trigonometric sum

Question

Where does the series $$ \sum_{k=1}^{\infty} { \frac{\sin(kx)}{k} } \space \space \space \text{for} \space x \in \mathbb{R}$$converge/ uniformly converge and how would I show this? I have tried many of the standard tests for convergence and got nowhere.

Answer 1

Use Dirichlet's criterion. Can you determine when $\displaystyle \sum_{k=1}^n \sin kx$ is (uniformly) bounded? Alternatively, a nice proof starts by observing $$\frac{\sin kx}k=\int_0^x \cos kt dt$$