TITLE: Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$
QUESTION [5 upvotes]: For any real $x$ and positive integer $n$, is it true that:
$$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$
Please justify.

REPLY [3 votes]: Looks like it. $|\sum_{k=1}^n\frac{\sin(kx)}k|$ is a partial sum of the Fourier series of the function $y=\frac{\pi-|x|}{2}$. Here are the sums for $n$=15 (gold) and $n$=1350 (green) along with the graph of $y=2\sqrt{\pi}$ (blue). Maybe this will help? The Gibbs effect seems to be the only thing that moves the sum anywhere towards $2\sqrt{\pi}$, but does so ever so slightly.