For questions about Cesàro summation and Cesàro summable sequences.
If $\{a_k\}_{k = 1}^{\infty}$ is a sequence, define a sequence of partial sums $s_k = a_1 + a_2 + ... + a_k$. We call the sequence $$\frac{s_k}{k} = \frac{a_1 + ... + a_k}{k}$$ the Cesàro means of the sequence.
The sequence $\{a_k\}$ is called Cesàro summable if the sequence of Cesàro means converges. If the original sequence $\{a_k\}$ converges to some $A$, then the sequence of Cesàro means also converges to $A$. But the converse does not hold: There are divergent sequences whose Cesàro means converge (e.g. $a_n = (-1)^n$).
Reference: Cesàro summation.
