Geometric series

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Revision as of 12:23, 10 January 2010 by imported>Peter Schmitt (remark on finite and infinite added)
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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.

Remark
Since every finite geometric sequence is the initial segment of a uniquely determined infinite geometric sequence every finite geometric series is the initial segment of a corresponding infinite geometric series. Therefore, while in elementary mathematics the difference between "finite" and "infinite" may be stressed, in mathematical texts "geometrical series" usually refers to the infinite series.

Examples

Positive ratio   Negative ratio
The series

and corresponding sequence of partial sums

is a geometric series with quotient

and first term

and therefore its sum is

  The series

and corresponding sequence of partial sums

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.)