In mathematics, an infinite geometric series of the form

${\displaystyle \sum _{n=1}^{\infty }ar^{n-1}=a+ar+ar^{2}+ar^{3}+\cdots }$

is divergent if and only if | r | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case

${\displaystyle \sum _{n=1}^{\infty }ar^{n-1}={\frac {a}{1-r}}.}$

This is true of any summation method that possesses the properties of regularity, linearity, and stability.

Examples

In increasing order of difficulty to sum:

• 1 − 1 + 1 − 1 + · · ·, whose common ratio is −1
• 1 − 2 + 4 − 8 + · · ·, whose common ratio is −2
• 1 + 2 + 4 + 8 + · · ·, whose common ratio is 2
• 1 + 1 + 1 + 1 + · · ·, whose common ratio is 1.

Motivation for study

It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method sums Σzⁿ to 1/(1 - z) for all z in a subset S of the complex plane, given certain restrictions on S, then the method also gives the analytic continuation of any other function f(z) = Σa_nzⁿ on the intersection of S with the Mittag-Leffler star for f.^[1]

Summability by region

Notes

References