Alternating integers have no first-order Cesàro sum ($1-2+3-4+\cdots \text{ is not }(C,1)\text{-summable}$)

For $a_n=(-1{)}^n(n+1)$, direct finite summation gives $s_{2m}=m+1$ and $s_{2m+1}=-(m+1)$. ConsequentlyThe 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.