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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?

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up vote 6 down vote accepted

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.

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Alternative ending: (not quite as quick to finish by more elementary): Use summation by parts. – Ragib Zaman Oct 31 '11 at 5:54
Typo: "but more elementary" that should have been. – Ragib Zaman Oct 31 '11 at 13:24
with summation by parts, what can i do? – August Nov 17 '11 at 2:03

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