 Alternating integers have no first-order Cesàro sum

ID: alternating-integers-have-no-first-order-cesaro-sum

For

a_{n} = (- 1)^{n} (n + 1)

, direct finite summation gives

s_{2 m} = m + 1

and

s_{2 m + 1} = - (m + 1)

. Consequently

σ_{2 m} = \frac{m + 1}{2 m + 1} ⟶ \frac{1}{2}, σ_{2 m + 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.

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