Cesaro summable

Let $(a_n{)}_{n=1}^{\infty }$ be a sequence and let

$$s_k=a_1+\dots +a_k=\sum _{n=1}^k{a}_n$$

be a the $k$th partial sum.

The sequence $a_n$ is Cesaro summable, with Cesaro sum $A\in \mathbb{R}$, if, as $n\to \infty$, the mean of the first $n$ partial sums $s_1,s_2,\dots ,s_n$ tends to $A$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=1}^n{s}_k=A$$