In mathematics, an infinite geometric series of the form
infty | |
\sum | |
n=1 |
arn-1=a+ar+ar2+ar3+ …
infty | |
\sum | |
n=1 |
arn-1=
a | |
1-r |
.
In increasing order of difficulty to sum:
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 on the intersection of S with the Mittag-Leffler star for f.[1]
Ordinary summation succeeds only for common ratios |z| < 1.
The series is Borel summable for every z with real part < 1. Any such series is also summable by the generalized Euler method (E, a) for appropriate a.
Certain moment constant methods besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 - z), that is, for all z except the ray z ≥ 1.[2]