# Must a series converge after a finite number of 'césaro mean' applications if it does after infinitely many.

Mathematics Asked by Oddly asymmetric on October 14, 2020

Let $$alpha=(alpha_n)_{nge0}$$ be a sequence of real numbers. Let $$C(alpha)=(beta_n)_{nge0}$$ where
$$beta_n=frac{1}{n+1}sum_{k=0}^{n}alpha_k$$
Finally let $$C^n=Ccirc C^{n-1}$$ with $$C^0=text{Id}$$ be functions from the set of real sequences to itself.

Will the pointwise limit $$C^{infty}(alpha)=gamma$$ always exist? Will it always be convergent (i.e. will $$lim_{ntoinfty}gamma_n$$ exist)?

I also want to know, if the pointwise limit sequence $$C^{infty}(alpha)$$ does exist and is convergent, must there be a finite n such that the sequence $$C^n(alpha)$$ is convergent? Certainly conversely if this sequence is convergent at a finite n then it is convergent for all greater values of n and so if the pointwise limit exists its limit will be the same value.

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