Near-periodic Monotonic Divergent Series of Integers

HYPERSPECIES

j-spaced and k-shifted monotonic divergent series of 1

$\displaystyle \sum_{\substack{n\geq m \\ j \text{ spacing} \\ k \text{ shifts}}} 1 = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = -\dfrac{1}{2} - \dfrac{k-j}{j+1} \qquad (m \in \mathbb{Z})$ (unstable)

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