5
$\begingroup$

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}? $

1 Answers 1

9

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.