For what complex values $z$ make the series $\sum_{n=0}^{\infty}\frac{2}{1+z^{n}}$ converge?

Question

This is my guess:

Since $z^{n}$ converges when $|z|<1$, the series in question must converge when $|z|>1$, since then $1>\frac{1}{|z|}$ and since $\frac{2}{1+z^{n}}<\frac{2}{z^{n}}$, the series converges by the comparison test.

Is this guess right though? Or am I wrong in some arguments?