Aperiodic Monotonic Divergent Series of Rational Numbers

SUBSPECIES

$\displaystyle \sum_{n\geq2} \dfrac{1}{n} = \zeta(1) - 1 = \gamma - 1$ (semi-stable)

$\displaystyle \sum_{n\geq2} \dfrac{1}{2n} = \dfrac{\gamma}{2} - \dfrac{1}{2}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{2n+1} = \sum_{n\geq2} \dfrac{1}{2n-1} = \dfrac{1}{2} \big(\gamma-H_{\frac{1}{2}}\big) = \dfrac{\gamma}{2} - 1 + \ln2$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{4n+1} = \sum_{n\geq2} \dfrac{1}{4n-3} = \dfrac{1}{4} \big(\gamma-H_{\frac{1}{4}}\big) = \dfrac{\gamma}{4} - 1 + \dfrac{\pi}{8} + \dfrac{3\ln2}{4}$ (semi-stable)

$\displaystyle \sum_{n\geq1} \dfrac{1}{4n+3} = \sum_{n\geq2} \dfrac{1}{4n-1} = \dfrac{1}{4} \big(\gamma-H_{\frac{3}{4}}\big) = \dfrac{\gamma}{4} - \dfrac{1}{3} - \dfrac{\pi}{8} + \dfrac{3\ln2}{4}$ (semi-stable)

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

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

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