Cesàro mean
Definition
Let ${\{{a}_{n}\}}_{n=0}^{\mathrm{\infty}}$ be a sequence of real (or possibly complex numbers^{}). The Cesàro mean of the sequence $\{{a}_{n}\}$ is the sequence ${\{{b}_{n}\}}_{n=0}^{\mathrm{\infty}}$ 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  20130322 12:29:54 
Last modified on  20130322 12:29:54 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  11 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 4000 
Classification  msc 40G05 
Synonym  Cesaro mean 
Related topic  Sequence 
Related topic  CesaroSummability 
Related topic  StolzCesaroTheorem 