I have to figure out if the following series converge or diverge:
(1) $\sum_{n=1}^{\infty}\frac{(-i)^{n+1}}{n^{2}+1}$. This has the $N$-th partial sum $s_N=\sum^{N}\frac{(-i)^{N+1}}{N^{2}+1}$.
My attempt: To show whether or not the series converges, I need to show whether {$S_{N}$} is Cauchy. So I have to see that $\forall\epsilon>0$ $\exists N_{\epsilon}$ s.t. for all $n,m>N_{\epsilon}$, $|\sum_{k=m+1}^{n}\frac{(-i)^{n+1}}{n^{2}+1}|<\epsilon$. Where do I go from here?
(2) $\sum_{n=1}^{\infty}e^{in\theta}/n$ for $0<\theta<2\pi$, $\theta$ fixed.
Not sure how to approach this.