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# Cesàro summability

Cesàro summability is a generalized convergence criterion for infinite series. We say that a series $\sum_{{n=0}}^{\infty}a_{n}$ 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 $\sum_{{n=0}}^{\infty}a_{n}$ Cesàro converges to a limit $L$, if

$\frac{1}{N+1}(s_{0}+\ldots+s_{N})\rightarrow L\quad\text{as}\quad N\rightarrow\infty.$ |

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

###### Proposition 1.

Suppose that

$\sum_{{n=0}}^{\infty}a_{n}=L,$ |

in the usual sense that $s_{N}\rightarrow L$ as $N\rightarrow\infty$. 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

$\sum_{{n=0}}^{\infty}(-1)^{n}.$ |

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

$\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6},\ldots$ |

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^{1}^{1}This 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

$\frac{1}{N+1}(s_{0}+\ldots+s_{N})\rightarrow L\quad\text{as}\quad N\rightarrow\infty.$ |

Then,

$f(r)=\sum_{{n=0}}^{\infty}a_{n}r^{n}\rightarrow L\quad\text{as}\quad r% \rightarrow 1^{{-}}$ |

as well.

## Mathematics Subject Classification

40G05*no label found*

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