Geometric series

From Citizendium
Revision as of 05:24, 10 January 2010 by imported>Peter Schmitt (example added)
Jump to navigation Jump to search
This article is developed but not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable, developed Main Article is subject to a disclaimer.

A geometric series is a series associated with an infinite geometric sequence, i.e., the quotient q of two consecutive terms is the same for each pair.

A geometric series converges if and only if |q|<1.

Then its sum is where a is the first term of the series.

Example

The series

is a geometric series with quotient

and first term

and therefore its sum is

Power series

Any geometric series

can be written as

where

The partial sums of the power series Σxk are

because

Since

it is

and the geometric series converges for |x|<1 with the sum

and diverges for |x| ≥ 1.