Monotonic Divergent Series of zeta Function

HYPERSPECIES

$\displaystyle \sum_{\substack{n\geq0 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(n) = \gamma - 2 - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(n) = \gamma - \dfrac{1}{2} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq2 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(n) = \dfrac{1}{2} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq0 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(2n) = -\dfrac{5}{4} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(2n) = \dfrac{1}{4} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq2 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(2n) = \dfrac{1}{4} - \dfrac{\pi^2}{6} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq0 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(2n+1) = \gamma - \dfrac{5}{4} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(2n+1) = -\dfrac{1}{4} - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq2 \\ j \text{ spacing} \\ k \text{ shifts}}} \zeta(2n+1) = -\dfrac{1}{4} - \zeta(3) - \dfrac{k-j}{j+1}$ (unstable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{\zeta(n)}{n} = \gamma - \ln(j+1)$ (semi-stable)

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