Geometric series: Difference between revisions
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imported>Peter Schmitt (→Power series: edit continued) |
imported>Peter Schmitt (include complex numbers) |
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i.e., the quotient ''q'' of two consecutive terms is the same for each pair. | i.e., the quotient ''q'' of two consecutive terms is the same for each pair. | ||
A geometric series converges if and only if − | A geometric series converges if and only if − |''q''|<1. | ||
Its sum is <math> a \over 1-q </math> where ''a'' is the first term of series. | Its sum is <math> a \over 1-q </math> where ''a'' is the first term of the series. | ||
== Power series == | == Power series == |
Revision as of 18:27, 9 January 2010
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 are
because
Since
there is
and the geometric series converges for |x|<1 with the sum
and diverges for |x| ≥ 1.