= Alternating integers have no first-order Cesàro sum
{title2=$1-2+3-4+\cdots\text{ is not }(C,1)\text{-summable}$}
For $a_n=(-1)^n(n+1)$, direct finite <summation> gives $s_{2m}=m+1$ and $s_{2m+1}=-(m+1)$. Consequently
$$
\sigma_{2m}=\frac{m+1}{2m+1}\longrightarrow\frac12,\qquad
\sigma_{2m+1}=0.
$$
The distinct even and odd <subsequential limits> prevent first-order <Cesàro summation>. More averaging is a different method; a value assigned by another method is not a first-order <Cesàro sum>.
Back to article page
