Prove this series is divergent: $\sum_{k=1}^{\infty}\sin kx$

Question

I need to prove that $\sum_{k=1}^{\infty}\sin kx$ is divergent when $x \notin \pi \Bbb Z$.

I tried to solve this equation with it's sums' seria but I didn't succeed. I'll be glad for some help.

Answer 1

HINT: (Edited incorporating Carl's comments)

$\sin kx $ doesn't tend to $0$ as $k\rightarrow \infty$, for $x \notin \mathbb{Z} $.

Answer 2

Using the divergence test that is; if $\sum {{a_k}} $ converges then $a_k \to 0$, however in our case $\sin(kx)$ does not converges to $0$, e.g. take $x = \frac{\pi}{2}$.