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?

Solution 1:

$|1+z^n| \geq |z^n| - 1 = |z|^n - 1$ 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 $n \to \infty$.

Solution 2:

The first statement is a version of the triangle inequality.

$$\begin{align} |x|&=|x-y+y|\le|x-y|+|y|\\ &\implies |x|-|y|\le|x-y|. \end{align}$$

Similarly $|y|-|x|\le|x-y|$. Hence, $$||x|-|y||\le|x-y|.$$

The other inequality is equivalent to $$|z|^n\le 2|z|^n-2$$ or $$|z|^n\ge 2.$$