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. 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 CesaroMean 2013-03-22 12:29:54 2013-03-22 12:29:54 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 40-00 msc 40G05 Cesaro mean Sequence CesaroSummability StolzCesaroTheorem