I'm trying to prove that, if $\displaystyle |z| > 1$ for any complex $\displaystyle z$, then $\displaystyle \displaystyle \sum^{\infty}_{n = 1} \frac{1}{1 + z^{n}}$ converges. I got this far but no farther:

If we try by comparison test, $\displaystyle |\frac{1}{1 + z^{n}}| \leq \frac{1}{|z|^{n} - 1} = \frac{1}{|z|^{n}} \cdot \frac{1}{1 - \frac{1}{|z|^{n}}}$. I know the first part of the RHS is decreasing, limit goes to 0. I'll be done if I show that the second part has bounded partial sums, but I can't figure that part out.

Thanks.