Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer 1

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.

share|improve this answer
1  
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.