TITLE: Complex number inequality?
QUESTION [5 upvotes]: Suppose $|z|>1$ for $z$ a complex number. I'm trying to build a certain comparison test to test convergence. I'm wondering, is it true that 
$$
\frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1}?
$$

REPLY [9 votes]: Yes. By triangle inequality, we have
$$|z|^n=|z^n|=|z^n+1-1|\leq|z^n+1|+1.$$
This implies that 
$$|z|^n-1\leq |z^n+1|.$$
Since $|z|>1$, we have $0<|z|^n-1\leq |z^n+1|$. Therefore, 
$$\frac{1}{|1+z^n|}\leq\frac{1}{|z|^n-1},$$
as required.