A Characterization of Cesaro Convergence

TL;DR

This paper characterizes Cesàro convergence of bounded sequences through the convergence of averages over specific index intervals, providing a new criterion for Cesàro convergence and divergence.

Contribution

It introduces a novel characterization of Cesàro convergence using averages over exponential index intervals, extending understanding of convergence criteria.

Findings

Cesàro convergence is equivalent to convergence of averages over intervals [α^k, α^{k+1}) for all α>1.

Nonnegative sequences are Cesàro convergent to 0 if and only if the average condition holds for some α>1.

Provides a new criterion simplifying the analysis of Cesàro convergence for bounded sequences.

Abstract

We show that a real bounded sequence $(x_n)$ is Ces\`aro convergent to $\ell$ if and only if the sequence of averages with indices in $[\alpha^k,\alpha^{k+1})$ converges to $\ell$ for all $\alpha>1$. If, in addition, the sequence $(x_n)$ is nonnegative, then it is Ces\`aro convergent to $0$ if and only if the same condition holds for some $\alpha>1$.