Geometric series

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Revision as of 19:25, 9 January 2010 by imported>Peter Schmitt (→‎Power series: edit continued)
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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 −1<q<1.

Its sum is where a is the first term of series.

Power series

Any geometric series

can be written as

where

The partial sums of the power series are

because

Since

there is

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

and diverges for |x| ≥ 1.