Revision as of 05:35, 10 January 2010 by imported>Peter Schmitt
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
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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 (more precisely: converges absolutely) for |x|<1 with the sum

and diverges for |x| ≥ 1.
(It diverges definitely — to +∞ or −∞ depending on the sign of a — for x≥1.)