Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$ $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|>1$.
There are two facts that my professor uses that I am confused about.
The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any $|z|$.
The other is: $\frac{1}{|z|^n-1} \leq \frac{2}{|z|^n}$, I believe this is also true for any $|z|$.
Can anyone prove these statements for me?

$|1+z^n| \geq |z^n| - 1 = |z|^n - 1$ is true due to the $\triangle$ inequality, and the second one is true if $|z|^n \leq 2|z|^n - 2 \iff 2 \leq |z|^n$, and this is true for some $n \geq N_0$ since $|z|^n \to +\infty$ as $n \to \infty$.
