Divergent geometric series explained

In mathematics, an infinite geometric series of the form

	infty
\sum
	n=1

ar^n-1=a+ar+ar²+ar³+ …

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

	infty
\sum
	n=1

ar^n-1=

	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 assigns $\sum_^ z^n$ to

1/(1-z)

for all

z

in a subset

of the complex plane, given certain restrictions on

, then the method also gives the analytic continuation of any other function

f(z) = \sum_^ a_n z^n

on the intersection of

with the Mittag-Leffler star for

f(z)

.^[1]

Summability by region

Open unit disk

Ordinary summation succeeds only for common ratios

|r|<1.

Closed unit disk

Cesàro summation
Abel summation

Larger disks

Euler summation

Half-plane

The series is Borel summable for every z with real part < 1.

Shadowed plane

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]

Everywhere

References

Book: Korevaar, Jacob . Tauberian Theory: A Century of Developments . Springer . 2004 . 3-540-21058-X.
Alexander . Moroz . Quantum Field Theory as a Problem of Resummation . 1991 . hep-th/9206074 .

Notes and References

Korevaar p.288
Moroz p.21