First-order Cesàro summation of a geometric series ($(C,1){\sum }_{n=0}^{\infty }{z}^n=(1-z{)}^{-1}(|z|\le 1,z\ne 1)$)

For the geometric series with terms $z^n$, $z\ne 1$, the partial sums are $(1-z^{n+1})/(1-z)$. Averaging the finite expressions givesIf $|z|\le 1$ and $z\ne 1$, the second term has absolute value at most $2/((N+1)|1-z{|}^2)$, and therefore tends to zero. On the unit circle, except at $z=1$, this yields a first-order Cesàro sum even though the terms do not tend to zero. For $z=1$, instead $s_n=n+1$ and ${\sigma }_N=(N+2)/2$, so there is no finite Cesàro sum.^[ref]