Cesàro summability

Cesàro summability is a generalized convergence criterion for infinite series. We say that a series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ is Cesàro summable if the Cesàro means of the partial sums converge to some limit $L$. To be more precise, letting

$$s_N=\sum _{n=0}^N{a}_n$$

denote the $N^{\text{th}}$ partial sum, we say that ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ Cesàro converges to a limit $L$, if

$$\frac{1}{N+1}(s_0+\mathrm{\dots }+s_N)\to L\mathit{\hspace{1em}}\text{as}\mathit{\hspace{1em}}N\to \mathrm{\infty }.$$

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

Proposition 1

Suppose that

$$\sum _{n=0}^{\mathrm{\infty }}{a}_n=L,$$

in the usual sense that $s_N\mathrm{\to }L$ as $N\mathrm{\to }\mathrm{\infty }$. Then, the series in question Cesàro converges to the same limit.

The converse, however is false. The standard example of a divergent series, that is nonetheless Cesàro summable is

$$\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n.$$

The sequence of partial sums $1,0,1,0,\mathrm{\dots }$ does not converge. The Cesàro means, namely

$$\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6},\mathrm{\dots }$$

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

There is also a relation between Cesàro summability and Abel summability.

Theorem 2 (Frobenius)

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

$$\frac{1}{N+1}(s_0+\mathrm{\dots }+s_N)\to L\mathit{\hspace{1em}}\text{𝑎𝑠}\mathit{\hspace{1em}}N\to \mathrm{\infty }.$$

Then,

$$f(r)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{r}^n\to L\mathit{\hspace{1em}}\text{𝑎𝑠}\mathit{\hspace{1em}}r\to 1^{-}$$

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