TITLE: Proving the series of partial sums of $\sin (in)$ is bounded?
QUESTION [5 upvotes]: So I'm trying to prove some series converges, and I'm trying to show it by using the Drichlet test. 
So I need to prove that the series of partial sums $\Sigma_{k=1}^\infty \sin(ki)$ is bounded. I tried proving it by dividing and multiplying with $2\cos(\frac{i}{2})$ and then using the trigonometric identity $2\cos(\beta)\sin(\alpha)=sin(\alpha+\beta)-sin(\alpha-\beta)$, which creates a telescopic series - but I didn't manage to bound the result.
Any assistance would be great! Thanks in advance!

REPLY [4 votes]: One way to obtain a bound is to derive a closed-from expression for the sum.  Here, this is accomplished by summing  geometric series. To that end,
$$\begin{align}
\left|\sum_{n=1}^N \sin(nx)\right| &=\left| \text{Im} \left(\sum_{n=1}^N e^{inx}  \right) \right|\\
&=\left|\text{Im} \left( \frac{e^{ix}-e^{i(N+1)x}}{1-e^{ix}} \right)\right| \\
&=\left|\frac{\sin \left(\frac{Nx}{2} \right) \sin\left(\frac{(N+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)}\right|\\
&\le \left|\csc\left(\frac{x}{2}\right)\right|
\end{align}$$
Note:  A tighter bound can be obtained by noting 
$$\left|\sin \left(\frac{Nx}{2} \right) \sin\left(\frac{(N+1)x}{2}\right)\right|= \frac12 \left||\cos (\frac{x}{2})-\cos(N+\frac12)x \right|\le \frac12\left(1+\left|\cos (\frac{x}{2})\right|\right)$$