Cesàro summability

Cesàro summability is a generalized convergence criterion for infinite series. We say that a series n=0an is Cesàro summable if the Cesàro means of the partial sums convergePlanetmathPlanetmath to some limit L. To be more precise, letting


denote the Nth partial sum, we say that n=0an Cesàro converges to a limit L, if


Cesàro summability is a generalizationPlanetmathPlanetmath of the usual definition of the limit of an infinite series.

Proposition 1

Suppose that


in the usual sense that sNL as N. Then, the series in question Cesàro converges to the same limit.

The converseMathworldPlanetmath, however is false. The standard example of a divergent seriesMathworldPlanetmath, that is nonetheless Cesàro summable is


The sequence of partial sums 1,0,1,0, does not converge. The Cesàro means, namely


do converge, with 1/2 as the limit. Hence the series in question is Cesàro summable.

There is also a relationMathworldPlanetmath between Cesàro summability and Abel summability11This and similar results are often called Abelian theorems..

Theorem 2 (Frobenius)

A series that is Cesàro summable is also Abel summable. To be more precise, suppose that




as well.

Title Cesàro summability
Canonical name CesaroSummability
Date of creation 2013-03-22 13:07:01
Last modified on 2013-03-22 13:07:01
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 6
Author rmilson (146)
Entry type Definition
Classification msc 40G05
Related topic CesaroMean