Geometric series: Difference between revisions
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'''Remarks''' <br> | '''Remarks''' <br> | ||
# The sum of | # The sum of finite (''n'') terms of a geometric sequence is a finite number ''S''<sub>''n''</sub>; its formula is given below. | ||
#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 more advanced mathematical texts "geometrical series" usually refers to the infinite series. | #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 more advanced mathematical texts "geometrical series" usually refers to the infinite series. | ||
Revision as of 03:13, 11 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. Thus, the series has the form
where the quotient (ratio) of the (n+1)th and the nth term is
An infinite geometric series (i.e., a series with an infinite number of terms) converges if and only if |q|<1, in which case its sum is , where a is the first term of the series.
Remarks
- The sum of finite (n) terms of a geometric sequence is a finite number Sn; its formula is given below.
- 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 more advanced 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 |
The partial sum S5 follows thus (see the formula derived below)
Power series
By definition, a geometric series
can be written as
where
The partial sums of the power series Σqk are
because
Since
it is
and the geometric series converges (more precisely: converges absolutely) for |q|<1 with the sum
and diverges for |q| ≥ 1. (Depending on the sign of a, the limit is +∞ or −∞ for q≥1. In the case q≤-1 partial sums oscillate and hence the sign of the limit is undetermined.)