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 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

Open unit disk

Ordinary summation succeeds only for common ratios |z| < 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. Any such series is also summable by the generalized Euler method (E, a) for appropriate a.

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]

Notes

References

Sequences and series Arithmetic sequence Divergent series 1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ Infinite arithmetic series Geometric sequence Convergent series 1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ Divergent geometric series 1 + 1 + 1 + 1 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Power of two Power of 10 Hypergeometric series Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series Integer sequence Complete Cubic number Factorial Fibonacci numbers Figurate number Heptagonal number Hexagonal number List Lucas number Pell number Pentagonal number Polygonal number Square number Triangular number Other sequence Divergent series 1 − 2 + 3 − 4 + ⋯ 1 − 1 + 2 − 6 + 24 − 120 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ Periodic sequence