How to show this sum of exponentials is zero

Question

How do you show that $$\sum_{n=-\infty}^\infty e^{ikan}=0$$ for $a>0$ and $k$ not an integer multiple of $\frac{2\pi}{a}$? My first guess would be to look at the partial sum $$S_N=\sum_{n=-N}^Ne^{ikan}$$ and see if this converges when $N\rightarrow\infty$. But $S_N$ oscillates with constant amplitude as $N$ grows larger. For example for $k=0.1$ the value of $S_N$ oscillates between about $[-20,20]$ and since this oscillation doesn't seem to die down it appears as if this sum doesn't converge.

Answer 1

It is not true! It is $$S_N = \sum_{n=-N}^{N} e^{ikan} = 1+ 2\sum_{n=1}^{N}\left( \frac{e^{ikan}+e^{-iakn}}{2}\right) = 1+2\sum_{n=1}^{N}\cos(akn)$$ Now observe that $(\cos(akn))_{n\in \mathbb{N}}$ is no null sequence, so $S_N$ cannot converge.