Weighted partial sums for Cesàro summation
= Weighted partial sums for Cesàro summation
{title2=$\sigma_N=\sum_{k=0}^{N}(1-k/(N+1))a_k$}
In the average of <partial sums> $s_0,\ldots,s_N$, the term $a_k$ occurs exactly $N-k+1$ times. Reordering this finite <sum> gives the displayed identity. Early terms have weights closer to one, and the newest term has weight $1/(N+1)$. The triangular weight expresses the cumulative averaging rule; it is not a claim that one may freely rearrange an infinite <series>.