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# geometric series

A *geometric series* is a series of the form

$\displaystyle\sum_{{i=1}}^{n}ar^{{i-1}}$ |

(with $a$ and $r$ real or complex numbers). The partial sums of a geometric series are given by

$\displaystyle s_{n}=\sum_{{i=1}}^{n}ar^{{i-1}}=\frac{a(1-r^{n})}{1-r}.$ | (1) |

An *infinite geometric series* is a geometric series, as above, with $n\rightarrow\infty$. It is denoted by

$\displaystyle\sum_{{i=1}}^{\infty}ar^{{i-1}}$ |

$\displaystyle\sum_{{i=1}}^{\infty}ar^{{i-1}}=\frac{a}{1-r}$ | (2) |

Taking the limit of $s_{n}$ as $n\rightarrow\infty$, we see that $s_{n}$ diverges if $|r|\geq 1$. However, if $|r|<1$, $s_{n}$ approaches (2).

One way to prove (1) is to take

$\displaystyle s_{n}=a+ar+ar^{2}+\cdots+ar^{{n-1}}$ |

and multiply by $r$, to get

$\displaystyle rs_{n}=ar+ar^{2}+ar^{3}+\cdots+ar^{{n-1}}+ar^{{n}}$ |

subtracting the two removes most of the terms:

$\displaystyle s_{n}-rs_{n}=a-ar^{n}$ |

factoring and dividing gives us

$\displaystyle s_{n}=\frac{a(1-r^{n})}{1-r}$ |

$\square$

Defines:

infinite geometric series

Keywords:

infinite series

Related:

GeometricSequence, ExampleOfAnalyticContinuation, ApplicationOfCauchyCriterionForConvergence

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

40A05*no label found*

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## Corrections

suggestion by matte ✓

geometric series by perucho ✓

some formulas by matte ✓

s_n is not defined before it occurs by Mathprof ✓

extra "+" by pahio ✓

geometric series by perucho ✓

some formulas by matte ✓

s_n is not defined before it occurs by Mathprof ✓

extra "+" by pahio ✓