I am trying to investigate the behaviour of $\sum_{n=1}^{\infty} \frac{1}{1+z^n}$ where $z$ is a complex number. This post discussed how to prove it converges when $|z|>1$. But I am interested in a general discussion of its convergence.
My attempt is to use the ratio test:
The series $\sum a_n$ converges if $\lim_{n \rightarrow \infty} \sup \lvert \frac{a_{n+1}}{a_n} \rvert < 1$; diverges if $\lvert \frac{a_{n+1}}{a_n} \rvert \ge 1$ for all $n \ge n_0$, where $n_0$ is some fixed integer.
The challenge is to find a way to discuss the value of $\lvert \frac{a_{n+1}}{a_n} \rvert$ in a concise way. I tried to define $z = a+bi$ and use the binomial theorem to investigate the relation between $z_{n+1}$ and $z_n$. However, the argument easily becomes very complicated. I am wondering if there is a better approach to this problem.
