Near-periodic Alternate Divergent Series of Integers

HYPERSPECIES

j-spaced and k-shifted Grandi series [1-1]

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} (-1)^{n-1} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}-1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}-1+\underbrace{0+\cdots+0}_{j\text{ spacing}}\pm\cdots = \dfrac{1}{2}$ (stable)

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