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

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## 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:

\(\displaystyle \cmod {\dfrac z {1 - z} }\) | \(<\) | \(\displaystyle 1\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \cmod z^2\) | \(<\) | \(\displaystyle \cmod {1 - z}^2\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 + y^2\) | \(<\) | \(\displaystyle \paren {1 - x}^2 + y^2\) | Definition of Complex Modulus, where $z = x + i y$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 + y^2\) | \(<\) | \(\displaystyle 1 - 2 x + x^2 + y^2\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 0\) | \(<\) | \(\displaystyle 1 - 2 x\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 2 x\) | \(<\) | \(\displaystyle 1\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x\) | \(<\) | \(\displaystyle \dfrac 1 2\) |

$\Box$

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:

\(\displaystyle S'\) | \(=\) | \(\displaystyle \sum_{n \mathop = 1}^\infty w^n\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac w {1 - w}\) | Sum of Infinite Geometric Sequence: Corollary 1 |

The result follows.

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.3$. Series: Example $\text{(iii)}$