TITLE: proving that some series converges uniformly
QUESTION [5 upvotes]: How can I prove that the series $
\sum\limits_{n = 1}^\infty  {\frac{{\sin \left( {nx} \right)}}
{n}} 
$ converges uniformly on the interval $
[\varepsilon ,2\pi  - \varepsilon ]\,\,\varepsilon  > 0
$
In general , it´s difficult to me to prove that some sequence converges uniformly, for example this case, I can´t use the Weierstrass test here, there are some techniques to prove this kind of convergence?

REPLY [6 votes]: Since $\sum_{k=1}^n\sin kx=\frac{\sin(nx/2)\sin((n+1)x/2)}{\sin(x/2)}$ is bounded in $[\epsilon, 2\pi-\epsilon]$, you can use Dirichlet's Test for Uniform Convergence.