proving that some series converges uniformly

Question

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?

Answer 1

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.