Aperiodic Monotonic Divergent Series of Integers

HYPERSPECIES

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

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

j-spaced and k-shifted monotonic divergent series of natural numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} n = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+2+\underbrace{0+\cdots+0}_{j\text{ spacing}} \\ +3+\underbrace{0+\cdots+0}_{j\text{ spacing}}+4+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{-\dfrac{1}{12}}_{\text{seed}} + \dfrac{1}{2} \left(\dfrac{k-j}{j+1}\right)^2$ (unstable)

j-spaced and k-shifted monotonic divergent series of odd numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} (2n-1) = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+3+\underbrace{0+\cdots+0}_{j\text{ spacing}} \\ +5+\underbrace{0+\cdots+0}_{j\text{ spacing}}+7+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{1}{3}}_{\text{seed}} + \dfrac{(k+1)(k-j)}{(j+1)^2}$ (unstable)

j-spaced and k-shifted monotonic divergent series of triangular numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{n(n+1)}{2} = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+\underbrace{0+\cdots+0}_{j\text{ spacing}}+3+\underbrace{0+\cdots+0}_{j\text{ spacing}} \\ +6+\underbrace{0+\cdots+0}_{j\text{ spacing}}+10+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{-\dfrac{1}{24}}_{\text{seed}} + \dfrac{1}{12} \left(\dfrac{k-j}{j+1}\right)^2$ (unstable)

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