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Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's test. But I don't know how to judge whether it is continuous.

Could you tell me the answer and why? Thank you in advance!

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How do you show convergence by Dirichlet's test? For example if $x=1$ it isn't alternately positive then negative... – coffeemath Nov 14 '13 at 15:11
$1/n$ is decreasing and the partial sum of $\sum_{n=1}^{\infty}{sin(nx)}$ is bounded. So the sum of the product is convergent. – F.G Nov 14 '13 at 15:14
Yes, got it. Thanks, and +1 for an interesting question. – coffeemath Nov 14 '13 at 15:16
Is it somehow obvious that the partial sums of $\sum \sin(nx)$ are bounded? I'm not seeing it... – Jason DeVito Nov 14 '13 at 15:58
@JasonDeVito It is the imaginary part of a geometric series. – achille hui Nov 14 '13 at 16:00
up vote 11 down vote accepted

To inspect the discontinuity of the summation, let's calculate the sum. By the Abel's theorem,

$$ f(x) := \sum_{n=1}^{\infty} \frac{\sin nx}{n} = \lim_{s\to 0^{+}} \sum_{n=1}^{\infty} \frac{\sin nx}{n} e^{-ns}. $$

By utilizing Taylor expansion of the logarithm,

\begin{align*} \sum_{n=1}^{\infty} \frac{\sin nx}{n} e^{-ns} &= \Im \sum_{n=1}^{\infty} \frac{e^{n(ix-s)}}{n} = - \Im \log (1 - e^{ix-s}) \\ &= -\Im \log (1 - e^{-s}\cos x - ie^{-s}\sin x) \\ &= \arctan \left(\frac{e^{-s}\sin x}{1 - e^{-s}\cos x}\right). \end{align*}

Thus taking $s \to 0^{+},$

$$ f(x) = \arctan \left(\frac{\sin x}{1 - \cos x}\right) = \arctan \left(\cot \frac{x}{2}\right) = \arctan \left(\tan \frac{\pi-x}{2}\right). $$


$$ f(x) = \begin{cases} \frac{\pi - x}{2} & x \in (0, 2\pi),\\ 0 & x = 0, \\ f(x+2\pi), & x \in \Bbb{R}. \end{cases} $$

This shows a clear-cut jump discontinuity at each $x \in 2\pi \Bbb{Z}$.

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I rethink this question today.And I observe that if we drop out "$e^{-ns}$",the proof can still work,in other words, for $x\neq0$ \begin{align*} \sum_{n=1}^{\infty} \frac{\sin nx}{n} &= \Im \sum_{n=1}^{\infty} \frac{e^{nix}}{n} = - \Im \log (1 - e^{ix}) \\ &= -\Im \log (1 - \cos x - i\sin x) \\ &= \arctan \left(\frac{\sin x}{1 - \cos x}\right). \end{align*} And if $x=0$ ,the summation is $0$. – F.G Nov 28 '13 at 8:01
@F.G Tha ks for pointing out that. I was also aware of that, but I adopted this regularizing method in order to avoid possible problems arising from the boundary behavior and the convergence mode of the series. – Sangchul Lee Nov 28 '13 at 8:08

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