Divergent geometric series

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In mathematics, an infinite geometric series of the form

is divergent if and only if |r| ≥ 1. Often a summation method for divergent series will assign to certain divergent geometric series a generalized sum that agrees with the classical formula

a/(1 + r).

Contents 1 Examples 2 Motivation for study 3 Summability by region 3.1 Open unit disk 3.2 Closed unit disk 3.3 Larger disks 3.4 Half-plane 3.5 Shadowed plane 3.6 Everywhere 4 Notes 5 References

[edit] 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

[edit] 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]

[edit] Summability by region

[edit] Open unit disk

Ordinary summation succeeds only for common ratios |z| < 1.

[edit] Closed unit disk

• Cesàro summation
• Abel summation

[edit] Larger disks

• Euler summation

[edit] 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.

[edit] 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]

[edit] Everywhere

[edit] Notes

[edit] References