Divergent geometric series
In mathematics, an infinite geometric series of the form
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
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 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) = Σanzn on the intersection of S with the Mittag-Leffler star for f.[1]
Summability by region[]
Open unit disk[]
Ordinary summation succeeds only for common ratios |z| < 1.
Closed unit disk[]
- Cesàro summation
- Abel summation
Larger disks[]
Half-plane[]
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.
Shadowed plane[]
Certain 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]
Everywhere[]
Notes[]
References[]
- Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
- Moroz, Alexander (1991). "Quantum Field Theory as a Problem of Resummation". arXiv:hep-th/9206074.
- Divergent series
- Geometric series