Absolutely Convergent Complex Series/Examples/(z over (1-z))^n

Example of Absolutely Convergent Complex Series
The complex series defined as:
 * $\displaystyle S = \sum_{n \mathop = 1}^\infty \paren {\dfrac z {1 - z} }^n$

is absolutely convergent, provided $\Re \paren z < \dfrac 1 2$.

Proof
Suppose $S$ is absolutely convergent.

Then $\displaystyle \sum_{n \mathop = 1}^\infty \cmod {\dfrac z {1 - z} }^n$ is convergent.

By Terms in Convergent Series Converge to Zero, this means that:
 * $\lim_{n \mathop \to \infty} \cmod {\dfrac z {1 - z} }^n \to 0$

which means in turn that:

It remains to be shown that $S' := \displaystyle \sum_{n \mathop = 1}^\infty \cmod {\dfrac z {1 - z} }^n$ is in fact a convergent series when $z < \dfrac 1 2$.

When $z < \dfrac 1 2$, we have that $\cmod {\dfrac z {1 - z} } < 1$, from above.

Let $w = \cmod {\dfrac z {1 - z} }$.

Then we have that:

The result follows.