geometric series

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

The *geometric series* is the series

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

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.

See also:

- Wikipedia,
*Geometric series*