Geometric series

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Revision as of 20:25, 15 January 2010 by imported>Peter Schmitt (→‎Summary: Convergence behaviour of the geometric series: a few more details)
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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.

Thus, every geometric series has the form

where the quotient (ratio) of the (n+1)th and the nth term is

The sum of the first n terms of a geometric sequence is called the n-th partial sum (of the series); its formula is given below (Sn).

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.

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 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

Summary: Convergence behaviour of the geometric series

The geometric series

  • converges (more precisely: converges absolutely) for |q|<1 with the sum
  • and diverges for |q| ≥ 1.
  • For real q:
For q ≥ 1 the limit is +∞ or −∞ depending on the sign of a.
For q = −1 the series alternates between a and 0.
For q < −1 the sign of partial sums alternates, the limit of their absolute values is ∞, but no infinite limit exists.
  • For complex q:
For |q| = 1 and q ≠ 1 (i.e., q = −1 or non-real complex) the partial sums Sn are bounded but not convergent.
For |q| > 1 and q non-real complex the partial sums oscillate, the limit of their absolute values is ∞, but no infinite limit exists.