Cesàro summability

Cesàro summability is a generalized convergence criterion for infiniteseries. We say that a series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ isCesàro summable if the Cesàro means of the partial sums converge tosome 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 thelimit 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 inquestion Cesàro converges to the samelimit.

The converse, however is false. The standard example of a divergentseries, 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 inquestion is Cesàro summable.

There is also a relation between Cesàro summability and Abelsummability.

Theorem 2 (Frobenius)

A series that is Cesàro summable is also Abel summable. To be moreprecise, 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.