Let : $a_i \in [0,1]$, then does the arithmetic mean of the $a_i$ converge, so do we have :
$$\lim_{n \to \infty} \frac{\sum_{i=0}^n a_i}{n} \in \mathbb{R} $$
So far what I can say is that if the sequence $a_i$ converges then the A.M converges by Cesaro theorem.
Moreover all the $a_i$ are such that : $a_i = o(1/n)$, and the A.M is bounded by $1$ so maybe it does converge.
More generally what are the conditions on a positive sequence such that it A.M converges ?
Thank you !