Periodic Monotonic Divergent Series of Integers

SUPERSPECIES

Single-shifted monotonic divergent series of 1

$\displaystyle \sum_{n\geq1} \chi(n+1) = 0+1+1+1+1+1+1+1+\cdots = -\dfrac{3}{2}$ (unstable)

Single-spaced monotonic divergent series of 1

$\displaystyle \sum_{n\geq1} \dfrac{1-(-1)^n}{2} = 1+0+1+0+1+0+1+0+\cdots = 0$ (unstable)

Single-spaced and single-shifted monotonic divergent series of 1

$\displaystyle \sum_{n\geq1} \dfrac{1+(-1)^n}{2} = 0+1+0+1+0+1+0+1+\cdots = -\dfrac{1}{2}$ (unstable)

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