Geometric series: Difference between revisions
imported>Peter Schmitt (→Example: added analog example with negative ratio) |
imported>Peter Schmitt (→Examples: formatting: putting the two examples side by side) |
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== Examples == | == Examples == | ||
{| class="wikitable" | |||
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! Positive ratio | |||
The series | ! width=100px | | ||
! Negative ratio | |||
|- | |||
| The series | |||
: <math> 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} + \cdots </math> | : <math> 6 + 2 + \frac 2 3 + \frac 2 9 + \frac 2 {27} + \cdots </math> | ||
and corresponding sequence of partial sums | and corresponding sequence of partial sums | ||
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and therefore its sum is | and therefore its sum is | ||
: <math> { 6 \over 1-\frac 13 } = 9 </math> | : <math> { 6 \over 1-\frac 13 } = 9 </math> | ||
| | |||
| The series | |||
The series | |||
: <math> 6 - 2 + \frac 2 3 - \frac 2 9 + \frac 2 {27} -+ \cdots </math> | : <math> 6 - 2 + \frac 2 3 - \frac 2 9 + \frac 2 {27} -+ \cdots </math> | ||
and corresponding sequence of partial sums | and corresponding sequence of partial sums | ||
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and therefore its sum is | and therefore its sum is | ||
: <math> { 6 \over 1- \left( - \frac 13 \right) } = \frac 9 2 </math> | : <math> { 6 \over 1- \left( - \frac 13 \right) } = \frac 9 2 </math> | ||
|} | |||
== Power series == | == Power series == |
Revision as of 11:15, 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.
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.)