Aperiodic Alternate Divergent Series of Rational Numbers

HYPERSPECIES

j-spaced and k-shifted alternate divergent series of rational numbers

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} \dfrac{(-1)^{n-1}}{n} = \ln 2$ (convergent)

SUPERSPECIES

SPECIES

$\displaystyle \sum_{n\geq1} \dfrac{(-1)^{n-1}}{n} = \ln 2$ (convergent)

$\displaystyle \sum_{n\geq0} \dfrac{(-1)^n}{2n-1} = -1 - \dfrac{1}{2} \left(\ln 2 + H_{-\frac{1}{4}} - H_{-\frac{1}{2}}\right) = -1 - \dfrac{\pi}{4}$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{(-1)^{n-1}}{2n-1} = \dfrac{1}{2} \left(\ln 2 + H_{-\frac{1}{4}} - H_{-\frac{1}{2}}\right) = \dfrac{\pi}{4}$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{(-1)^{n-1}}{3n-2} = \dfrac{1}{3} \left(\ln 2 + H_{-\frac{1}{3}} - H_{-\frac{2}{3}}\right) = \dfrac{\pi}{3\sqrt{3}} + \dfrac{\ln 2}{3}$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{(-1)^{n-1}}{3n-1} = \dfrac{1}{3} \left(\ln 2 + H_{-\frac{1}{6}} - H_{-\frac{1}{3}}\right) = \dfrac{\pi}{3\sqrt{3}} - \dfrac{\ln 2}{3}$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{(-1)^{n-1}}{n^2-1} = -\dfrac{3}{4}$ (convergent)

$\displaystyle \sum_{n\geq1} \dfrac{(-1)^{n-1}}{(2n-1)^2} = G = \beta(2) = \dfrac{\psi_1(\frac{1}{4})}{8} - \dfrac{\pi^2}{8}$ (convergent)

SUBSPECIES

$\displaystyle \sum_{n\geq2} \dfrac{(-1)^n}{n} = 1 - \ln 2$ (convergent)

$\displaystyle \sum_{n\geq2} \dfrac{(-1)^n}{2n-1} = 1 - \dfrac{\pi}{4}$ (convergent)

$\displaystyle \sum_{n\geq2} \dfrac{(-1)^n}{n^2-1} = \dfrac{1}{4}$ (convergent)

INFRASPECIES

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