Geometric series: Difference between revisions
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imported>Peter Schmitt m (→Power series: edit) |
imported>Peter Schmitt (→Power series: a_k are complex numbers) |
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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
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.