I'm trying to prove that this series converges, although I'm not entirely convinced that it does:
$\sum_{\substack{k=-\infty\\k\neq0}}^{\infty}\frac{e^{\frac{-\pi ik}{5}} - 1}{k}$
This is a two-sided infinite series of Fourier coefficients for a real valued 1-periodic function on $\mathbb{R}$, if that makes any difference. I haven't been able to find a way, does anyone have any ideas? Thanks.
Edit: Maybe the best way would be to somehow adapt the fact that the sum of the nth roots of unity is equal to zero..