Geometric series: Difference between revisions

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imported>Peter Schmitt
imported>Peter Schmitt
(→‎Power series: a_k are complex numbers)
Line 11: Line 11:


Any geometric series  
Any geometric series  
: <math> \sum_{k=1}^\infty a_k </math>
: <math> \sum_{k=1}^\infty a_k \qquad a_k \in \mathbb C </math>
can be written as
can be written as
: <math> a \sum_{k=0}^\infty x^k </math>
: <math> a \sum_{k=0}^\infty x^k </math>

Revision as of 18:52, 9 January 2010

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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 |q|<1.

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

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.