Cesàro mean

Definition

Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence of real (or possibly complex numbers). The Cesàro mean of the sequence $\{a_{n}\}$ is the sequence $\{b_{n}\}_{n=0}^{\infty}$ with

$b_{n}=\frac{1}{n+1}\sum_{i=0}^{n}a_{i}.$

(1)

0.0.1 Properties

1.

A key property of the Cesàro mean is that it has the same limit as the original sequence (when this limit exists). In other words, if $\{a_{n}\}$ and $\{b_{n}\}$ are as above, and $a_{n}\to a$ , then $b_{n}\to a$ . In particular, if $\{a_{n}\}$ converges, then $\{b_{n}\}$ converges too.

Title	Cesàro mean
Canonical name	CesaroMean
Date of creation	2013-03-22 12:29:54
Last modified on	2013-03-22 12:29:54
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	11
Author	mathcam (2727)
Entry type	Definition
Classification	msc 40-00
Classification	msc 40G05
Synonym	Cesaro mean
Related topic	Sequence
Related topic	CesaroSummability
Related topic	StolzCesaroTheorem