A Characterization of Cesaro Convergence

Abstract

We show that a real bounded sequence $(x_n)$ is Cesàro 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 Cesàro convergent to $0$ if and only if the same condition holds for some $\alpha >1$.

AMM, to appear