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 theoriginal 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.