#
nLab

geometric series

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.

## References

See also:

Last revised on November 9, 2020 at 08:46:34.
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