nLab geometric series

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.

Related entries

• series, power series

• Mercator series

References

See also:

• Wikipedia, Geometric series