---
tag: 01X5CVV1Q5
kind: example
name: Geometric series
msc: [40A05]
instance_of: [0AF7X08PCR]
uses: [0MNM54WB99, 0K0H5E3Z5N, 05APWSX53S, 0Q7K4AXE53]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_series
    locator: Infinite geometric series and convergence
same_as:
  wikidata: Q1306887
  wikipedia: Geometric series
---

## Example

Let $a$ and $r$ be real numbers with $|r| < 1$.
The geometric series $\sum_{n=0}^{\infty} a r^n$ converges, and
$$\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}.$$
The [[sum of a finite geometric series|05APWSX53S]] gives the partial sums $s_N = a (1 - r^{N+1})/(1 - r)$.
Since [[$r^{N+1} \to 0$|0Q7K4AXE53]], the partial sums tend to $a/(1 - r)$.

## Remarks

For $a \ne 0$ and $|r| \ge 1$, the series diverges by the [[term test|0VVY4H92G2]], because $|a r^n| \ge |a|$.
For $a = 1$ and $r = 1/2$, it gives $1 + 1/2 + 1/4 + \cdots = 2$.
The decimal $0.333\ldots$ is the geometric series with $a = 3/10$ and $r = 1/10$, and its sum is $1/3$.
The terms form a [[geometric sequence|0MNM54WB99]].

## Explanations

### L1 human

Walk half of the way to a wall, then half of the rest, and so on.
The distances $1/2 + 1/4 + 1/8 + \cdots$ add up to the whole distance $1$.