For which positive integer $k$ does the series $\sum_{n=1}^\infty\frac{\sin(n\pi/k)}{n}$ converge? 
For which positive integer $k$ does the series $$\sum_{n=1}^\infty\frac{\sin(n\pi/k)}{n}$$ converge?

The cases when $k=1$ or $k=2$ are trivial. For $k>2$, I don't see how to approach it. If one looks at the partial sums:
$$
s_k=\frac{\sin(\frac{\pi}{k})}{1}+\frac{\sin(2\cdot\frac{\pi}{k})}{2}+\cdots+\frac{\sin((k-1)\cdot\frac{\pi}{k})}{k-1}
$$
$$
s_{2k}-s_k=
(-1)\frac{\sin(\frac{\pi}{k})}{k+1}+(-1)\frac{\sin(2\cdot\frac{\pi}{k})}{k+2}+\cdots+(-1)\frac{\sin((k-1)\cdot\frac{\pi}{k})}{k+k-1}
$$
$$
s_{3k}-s_{2k}=\frac{\sin(\frac{\pi}{k})}{2k+1}+\frac{\sin(2\cdot\frac{\pi}{k})}{2k+2}+\cdots+\frac{\sin((k-1)\cdot\frac{\pi}{k})}{2k+k-1}\\
\vdots
$$
one might tend to sum the terms vertically so that the alternating test can be used. But unfortunately this series does not converge absolutely. 

$$\sum_{n=1}^\infty\frac{\sin (n\pi/k)}{n} = \sum_{n=1}^\infty\frac{\pi}{n\pi}\sin2\pi \frac{n}{2}\frac1{k}$$
The right hand side is a sawtooth wave of period $2$, at $t = 1/k$:
$$f(t) = \frac{\pi}{2}-\frac{\pi}2t, \quad 0<t<2\\
f(t+2) = f(t)\\
f(2n) = 0, \quad n\in\mathbb Z$$
So specifically,
$$\sum_{n=1}^\infty\frac{\sin (n\pi/k)}{n} = f\left(\frac1k\right) = \frac{\pi}{2} - \frac{\pi}{2k}$$

The Fourier series coefficients of $f(t)$ is given by
$$\begin{align*}
b_n &= \frac22\int_{0}^2\frac{\pi}{2}(1-t)\sin\frac{2\pi n t}{2}dt\\
&= \frac\pi2\int_0^2\sin\frac{2\pi n t}{2}dt - \frac\pi2\int_0^2t\sin\frac{2\pi n t}2dt\\
&= 0 + \frac1{2n}\int_0^2 t\ d\cos\frac{2\pi n t}{2}\\
&= \frac1{2n}\left[t\cos\frac{2\pi n t}2\right]_0^2 + \int_0^2\cos\frac{2\pi nt}2 dt\\
&= \left(\frac1{2n}\cdot2\cos2\pi n-0 \right) + 0\\
&= \frac1n
\end{align*}$$
