Geometric series

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Revision as of 10:50, 10 January 2010 by imported>Peter Schmitt (do not exclude "finite", use ratio)
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A geometric series is a series associated with a geometric sequence, i.e., the ratio (or quotient) q of two consecutive terms is the same for each pair.

An infinite 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 (more precisely: converges absolutely) for |x|<1 with the sum

and diverges for |x| ≥ 1. (Depending on the sign of a, the limit is +∞ or −∞ for x≥1.)