geometric series
A geometric series is a series of the form
n∑i=1ari-1 |
(with a and r real or complex numbers). The partial sums of a geometric series are given by
sn=n∑i=1ari-1=a(1-rn)1-r. | (1) |
An infinite geometric series is a geometric series, as above, with n→∞. It is denoted by
∞∑i=1ari-1 |
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
sn=a+ar+ar2+⋯+arn-1 |
and multiply by r, to get
rsn=ar+ar2+ar3+⋯+arn-1+arn |
subtracting the two removes most of the terms:
sn-rsn=a-arn |
factoring and dividing gives us
sn=a(1-rn)1-r |
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Title | geometric series |
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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 |