Abel summability

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Abel.htmlAbel summability is a generalized convergence criterion for power series. It extends the usual definition of the sum of a series, and gives a way of summing up certain divergent seriesMathworldPlanetmath. Let us start with a series n=0an, convergentMathworldPlanetmath or not, and use that series to define a power series


Note that for |r|<1 the summability of f(r) is easier to achieve than the summability of the original series. Starting with this observation we say that the series an is Abel summable if the defining series for f(r) is convergent for all |r|<1, and if f(r) converges to some limit L as r1-. If this is so, we shall say that an Abel converges to L.

Of course it is important to ask whether an ordinary convergent seriesMathworldPlanetmath is also Abel summable, and whether it converges to the same limit? This is true, and the result is known as Abel’s limit theorem, or simply as Abel’s theorem.

Theorem 1 (Abel)

Let n=0an be a series; let


denote the corresponding partial sums; and let f(r) be the corresponding power series defined as above. If an is convergent, in the usual sense that the sN converge to some limit L as N, then the series is also Abel summable and f(r)L as r1-.

The standard example of a divergent series that is nonetheless Abel summable is the alternating seriesMathworldPlanetmath


The corresponding power series is




this otherwise divergent series Abel converges to 12.

Abel’s theorem is the prototype for a number of other theorems about convergence, which are collectively known in analysis as Abelian theorems. An important class of associated results are the so-called Tauberian theorems. These describe various convergence criteria, and sometimes provide partial converses for the various Abelian theorems.

The general converse to Abel’s theorem is false, as the example above illustrates11We want the converse to be false; the whole idea is to describe a method of summing certain divergent series!. However, in the 1890’s http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Tauber.htmlTauber proved the following partial converse.

Theorem 2 (Tauber)

Suppose that an is an Abel summable series and that nan0 as n. Then, nan is convergent in the ordinary sense as well.

The proof of the above theorem is not hard, but the same cannot be said of the more general Tauberian theorems. The more famous of these are due to Hardy, Hardy-Littlewood, Weiner, and Ikehara. In all cases, the conclusion is that a certain series or a certain integral is convergent. However, the proofs are lengthy and require sophisticated techniques. Ikehara’s theorem is especially noteworthy because it is used to prove the prime number theoremMathworldPlanetmath.

Title Abel summability
Canonical name AbelSummability
Date of creation 2013-03-22 13:07:03
Last modified on 2013-03-22 13:07:03
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 7
Author rmilson (146)
Entry type Definition
Classification msc 40G10
Related topic CesaroSummability
Related topic AbelsLimitTheorem
Defines Abelian theorem
Defines Tauberian theorem