Divergent geometric series

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In mathematics, an infinite geometric series of the form

\sum_{k=0}^\infty ar^k = a + ar + ar^2 + ar^3 +\cdots

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

\sum_{k=0}^\infty ar^k = \frac{a}{1-r}.

This is true of any summation method that possesses the properties of regularity, linearity, and stability.


In increasing order of difficulty to sum:

Motivation for study[edit]

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) = Σanzn on the intersection of S with the Mittag-Leffler star for f.[1]

Summability by region[edit]

Open unit disk[edit]

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

Closed unit disk[edit]

Larger disks[edit]


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[edit]

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]



  1. ^ Korevaar p.288
  2. ^ Moroz p.21


  • 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.