Geometric series: Difference between revisions
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imported>Peter Schmitt (→Power series: two remarks added) |
imported>Peter Schmitt (→Power series: replace "definitely") |
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: <math> \sum_{k=1}^\infty a_k = { a \over 1-q }</math> | : <math> \sum_{k=1}^\infty a_k = { a \over 1-q }</math> | ||
and diverges for |''x''| ≥ 1. | and diverges for |''x''| ≥ 1. | ||
( | (The limit is +∞ or −∞ depending on the sign of ''a'' for ''x''≥1.) |
Revision as of 07:16, 10 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.
Then its sum is where a is the first term of the series.
Example
The series
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. (The limit is +∞ or −∞ depending on the sign of a for x≥1.)