Cesaro mean ($\frac{1}{n}{\sum }_{k=1}^n{a}_k$)

The Cesaro mean of the first $n$ terms of a sequence is their arithmetic average. If $a_n\to L$, then its Cesaro means also tend to $L$.

If a sequence converges, its Cesaro means have the same limit. Given a small tolerance, split the average error into a fixed finite initial segment and a tail whose individual errors are below that tolerance. The first contribution is a fixed constant divided by $n$, while the second is bounded by the tolerance. The converse fails: the alternating sequence $(-1{)}^n$ has means tending to zero while its terms do not converge.