Contents

# Contents

## Definition

The geometric series is the series

$\sum_{n = 0}^\infty r^n \,.$

## Properties

For $r \in \mathbb{R}$ with ${\Vert r \Vert} \lt 1$ this sequence converges

$\underset{n \to \infty}{\lim} \underoverset{k = 0}{n}{\sum} r^k \;=\; \frac{1}{1 - r} \,.$

For $r \in \mathbb{Z}_p$ with $r = p$ the above sequences converges in the p-adic metric.