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Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$

From Stewart, we cannot find a calculus 2 easy way to prove this:


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marked as duplicate by Marvis, Austin Mohr, TMM, Cameron Buie, Arkamis Nov 2 '12 at 21:02

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What's $[n]$ suppose to be? perhaps the question has a typo? –  Jean-Sébastien Nov 2 '12 at 19:05
Kerry just means $\sin(n)$ –  user17762 Nov 2 '12 at 19:12
@BabakSorouh: But floor of a natural would be quite redundant. –  Hagen von Eitzen Nov 2 '12 at 19:12
Yeah floor or $n$ is just $n$, but this may just be it because sum of $\sin(n)/n$ is $1/2(\pi-1)$ –  Jean-Sébastien Nov 2 '12 at 19:16
Also look here… –  user17762 Nov 2 '12 at 19:25

1 Answer 1

$\sum\limits^{\infty}_{n=1}\dfrac{\sin{n\theta}}{n}$ is the Fourier series for $f(\theta)=\dfrac{\pi-\theta}{2}, \;\; 0<\theta<2\pi,$ and converges uniformly on every closed interval $[\alpha, \beta], \;\; 0<\alpha<\beta<2\pi$ by Dirichlet's test.

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