geometric series

A geometric seriesMathworldPlanetmath is a series of the form


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

sn=i=1nari-1=a(1-rn)1-r. (1)

An infinite geometric series is a geometric series, as above, with n. It is denoted by


If |r|1, the infinite geometric series diverges. Otherwise it converges to

i=1ari-1=a1-r (2)

Taking the limit of sn as n, we see that sn diverges if |r|1. However, if |r|<1, sn approaches (2).

One way to prove (1) is to take


and multiply by r, to get


subtracting the two removes most of the terms:


factoring and dividing gives us


Title geometric series
Canonical name GeometricSeries
Date of creation 2013-03-22 12:05:37
Last modified on 2013-03-22 12:05:37
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 16
Author mathcam (2727)
Entry type Definition
Classification msc 40A05
Related topic GeometricSequence
Related topic ExampleOfAnalyticContinuation
Related topic ApplicationOfCauchyCriterionForConvergence
Defines infinite geometric series