ThmDex – An index of mathematical definitions, results, and conjectures.

Necessary and sufficient conditions for convergence of geometric complex series

Formulation 0

Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then

(1)	\begin{equation} \sum_{n = 0}^{\infty} z^n \in \mathbb{C} \quad \iff \quad \|z\| < 1 \end{equation}
(2)	\begin{equation} \|z\| < 1 \quad \implies \quad \sum_{n = 0}^{\infty} z^n = \frac{1}{1 - z} \end{equation}

Formulation 1

Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then

(1)	\begin{equation} \lim_{N \to \infty} \sum_{n = 0}^N z^n \in \mathbb{C} \quad \iff \quad \|z\| < 1 \end{equation}
(2)	\begin{equation} \|z\| < 1 \quad \implies \quad \lim_{N \to \infty} \sum_{n = 0}^N z^n = \frac{1}{1 - z} \end{equation}

Formulation 2

Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then

(1)	\begin{equation} \sum_{n = 0}^{\infty} z^n \in \mathbb{C} \quad \iff \quad \|z\| \in [0, 1) \end{equation}
(2)	\begin{equation} \|z\| \in [0, 1) \quad \implies \quad \sum_{n = 0}^{\infty} z^n = \frac{1}{1 - z} \end{equation}