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So I came across this series $\sum_{n=1}^\infty \frac{\sin nx}{n}$ for $x \in [0,\pi]$. The problem is to find all $x$ for which the series is uniformly convergent. Now, I am struggling to answer that question. My understanding was that $0$ and $\pi$ are the only points. But I cannot prove it, can someone please give me any ideas/hints?

Thank you

Martin Sleziak
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Ilia
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  • I don't know what you mean by uniform convergence for a specific $x$, but I think for $x = \frac{\pi}{2}$, you get the sum $1-\frac{1}{3}+\frac{1}{5}-\dots$ which I think is $\frac{\pi}{4}$, so other $x$'s do work. – mathworker21 Dec 19 '16 at 04:05
  • Yeah, I saw the question stated as "for which values of x does the series converge uniformly". – Ilia Dec 19 '16 at 04:07
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    See http://math.stackexchange.com/questions/28830/does-sum-frac-sinnxn-converge-uniformly-for-all-x-in-0-2-pi – Omry Dec 19 '16 at 04:31
  • And also http://math.stackexchange.com/questions/566856/is-sum-n-1-infty-frac-sinnxn-continuous – Omry Dec 19 '16 at 04:32

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