Weighted partial sums for Cesàro summation (${\sigma }_N={\sum }_{k=0}^N(1-k/(N+1))a_k$)

In the average of partial sums $s_0,\dots ,s_N$, 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.