Monotonic Divergent Series of Real Numbers

HYPERSPECIES

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing} \\ k \text{ shifts}}} \ln n = \dfrac{\ln(2\pi)}{2}$ (stable)

$\displaystyle \sum_{\substack{n\geq1 \\ j \text{ spacing}}} \ln n = \dfrac{\ln(2\pi)}{2}$ (stable)

$\displaystyle \sum_{\substack{n\geq1 \\ k \text{ shifts}}} \ln n = \dfrac{\ln(2\pi)}{2}$ (stable)

SUPERSPECIES

$\displaystyle \sum_{n\geq0} \sqrt{n} = \zeta\left(\!-\frac{1}{2}\right)$ (stable)

SPECIES

$\displaystyle \sum_{n\geq1} \dfrac{1}{\sqrt{n}} = \zeta\left(\frac{1}{2}\right)$ (stable)

$\displaystyle \sum_{n\geq1} \sqrt{n} = \zeta\left(\!-\frac{1}{2}\right)$ (stable)

$\displaystyle \sum_{n\geq1} \ln n = -\zeta'(0) = \dfrac{\ln(2\pi)}{2}$ (stable)

$\displaystyle \sum_{n\geq1} (\ln n)^2 = \zeta''(0) = -\dfrac{\pi^2}{24} - \dfrac{\ln^2(2\pi)}{2} + \dfrac{\gamma^2}{2} + \gamma_1$ (stable)

$\displaystyle \sum_{n\geq1} (\ln n)^3 = -\zeta^{(3)}(0) = \dfrac{\pi^2}{8}\ln(2\pi) + \dfrac{\ln^3(2\pi)}{2} - \dfrac {3\ln(2\pi)}{2}\,\gamma^2 - \gamma^3 \\ + \zeta(3) - 3\gamma_1\ln(2\pi) - 3\gamma\gamma_1 - \dfrac{3}{2}\gamma_2$ (stable)

$\displaystyle \sum_{n\geq1} \dfrac{\ln n}{n} = -\zeta'(1) = \zeta(1) \left(\ln(2\pi) + \zeta(1) + \dfrac{\ln A_0}{\zeta(0)}\right) = -\gamma^2$ (stable)

$\displaystyle \sum_{n\geq1} \dfrac{(\ln n)^2}{n} = \zeta''(1)$ (stable)

$\displaystyle \sum_{n\geq1} \dfrac{(\ln n)^3}{n} = -\zeta^{(3)}(1)$ (stable)

$\displaystyle \sum_{n\geq1} \dfrac{\ln n}{n^2} = -\zeta'(2) = -\zeta(2) \left(\ln(2\pi) + \zeta(1) + \dfrac{\ln A_1}{\zeta(-1)}\right) \\ = -\dfrac{\pi^2}{6} \left(\ln(2\pi) + \gamma - 12\ln A\right)$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{\ln n}{n^3} = -\zeta'(3)$ (convergent)

$\displaystyle \sum_{n\geq1} \ln(2n) = \dfrac{\ln\pi}{2}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \ln(2n\!-\!1) = 0$ (semi-stable)

$\displaystyle \sum_{n\geq1} \ln(n^2\!-\!1) = \ln\pi - \gamma$ (semi-stable)

$\displaystyle \sum_{n\geq1} \ln(4n^2\!-\!1) = 0$ (semi-stable)

SUBSPECIES

$\displaystyle \sum_{n\geq2} \ln n = \dfrac{\ln(2\pi)}{2}$ (semi-stable)

$\displaystyle \sum_{n\geq2} \ln(2n-1) = \sum_{n\geq1} \ln(2n+1) = 0$ (semi-stable)

$\displaystyle \sum_{n\geq2} \ln(n^2-1) = \ln\pi$ (semi-stable)

$\displaystyle \sum_{n\geq2} \ln(4n^2-1) = -\ln3$ (semi-stable)

INFRASPECIES

$\displaystyle \sum_{n\geq m+1} \ln n = \sum_{n\geq 1} \ln(n+m) = \dfrac{\ln(2\pi)}{2} \,-\, \ln m! \\$ (semi-stable)

$\displaystyle \sum_{n\geq m+1} \ln(2n) = \sum_{n\geq 1} \ln(2(n+m)) = \dfrac{\ln\pi}{2} \,-\, \ln m!$ (unstable)

$\displaystyle \sum_{n\geq m+1} \ln(2n-1) = \sum_{n\geq 1} \ln(2n+2m-1) = \dfrac{\ln\pi}{2} \,-\, \ln\!\Gamma\!\left(m+\frac{1}{2}\right)$ (unstable)

$\displaystyle \sum_{n\geq m+1} \ln(n^2-1) = \ln\!\left(\dfrac{2\pi}{m(m+1)}\right) \,-\, 2\ln\!\Gamma(m)$ (semi-stable)

$\displaystyle \sum_{n\geq m+1} \ln(4n^2-1) = \ln\!\left(\dfrac{2\pi}{2m+1}\right) \,-\, 2\ln\!\Gamma\!\left(m+\dfrac{1}{2}\right)$ (unstable)

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