Geometric series: Difference between revisions
imported>Peter Schmitt (→Examples: formatting: putting the two examples side by side) |
imported>Peter Schmitt (remark on finite and infinite added) |
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Then its sum is <math> a \over 1-q </math> where ''a'' is the first term of the series. | Then its sum is <math> a \over 1-q </math> where ''a'' is the first term of the series. | ||
'''Remark''' <br> | |||
Since every finite geometric sequence is the initial segment of a uniquely determined infinite geometric sequence | |||
every finite geometric series is the initial segment of a corresponding infinite geometric series. | |||
Therefore, while in elementary mathematics the difference between "finite" and "infinite" may be stressed, | |||
in mathematical texts "geometrical series" usually refers to the infinite series. | |||
== Examples == | == Examples == |
Revision as of 11:23, 10 January 2010
A geometric series is a series associated with a geometric sequence, i.e., the ratio (or quotient) q of two consecutive terms is the same for each pair.
An infinite geometric series converges if and only if |q|<1.
Then its sum is where a is the first term of the series.
Remark
Since every finite geometric sequence is the initial segment of a uniquely determined infinite geometric sequence
every finite geometric series is the initial segment of a corresponding infinite geometric series.
Therefore, while in elementary mathematics the difference between "finite" and "infinite" may be stressed,
in mathematical texts "geometrical series" usually refers to the infinite series.
Examples
Positive ratio | Negative ratio | |
---|---|---|
The series
and corresponding sequence of partial sums is a geometric series with quotient and first term and therefore its sum is |
The series
and corresponding sequence of partial sums 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. (Depending on the sign of a, the limit is +∞ or −∞ for x≥1.)