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

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.