Geometric series: Difference between revisions

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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"
=== Positive ratio ===
|-
 
! 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
Line 22: Line 25:
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>
 
| &nbsp;
=== Negative ratio ===
| 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

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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.)