Monotonic Series of Bernoulli Numbers

SPECIES

$\displaystyle \sum_{n\geq0} B_n = \Psi^{(1)}(1) + B_1 - \dfrac{1}{2} = \zeta(2) + B_1 - \dfrac{1}{2} = \dfrac{\pi^2}{6} + B_1 - \dfrac{1}{2} \quad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} B_n^{+} = \Psi^{(1)}(1) + B_1^{+} - \dfrac{1}{2} = \zeta(2) = \dfrac{\pi^2}{6} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} B_n^{-} = \Psi^{(1)}(1) + B_1^{-} - \dfrac{1}{2} = \zeta(2) - 1 = \dfrac{\pi^2}{6} - 1 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} B_n = \Psi^{(1)}(1) + B_1 - \dfrac{3}{2} = \zeta(2) + B_1 - \dfrac{3}{2} = \dfrac{\pi^2}{6} + B_1 - \dfrac{3}{2} \quad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} B_n^{+} = \Psi^{(1)}(1) + B_1^{+} - \dfrac{3}{2} = \zeta(2) - 1 = \dfrac{\pi^2}{6} - 1 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} B_n^{-} = \Psi^{(1)}(1) + B_1^{-} - \dfrac{3}{2} = \zeta(2) - 2 = \dfrac{\pi^2}{6} - 2 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq2} B_n = \sum_{n\geq1} B_{2n} = \Psi^{(1)}(1) + \zeta(0) - B_0 = \zeta(2) + \zeta(0) - B_0 \\ = \zeta(2) + 3\,\zeta(0) = \dfrac{\pi^2}{6} - \dfrac{3}{2} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} B_{2n} = \Psi^{(1)}(1) + \zeta(0) = \zeta(2) + \zeta(0) = \dfrac{\pi^2}{6} - \dfrac{1}{2} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} B_{2n+1} = B_1 \qquad \textit{(finite series)}$

$\displaystyle \sum_{n\geq0} B_{2n+1}^{+} = B_1^{+} = \dfrac{1}{2} \qquad \textit{(finite series)}$

$\displaystyle \sum_{n\geq0} B_{2n+1}^{-} = B_1^{-} = -\dfrac{1}{2} \qquad \textit{(finite series)}$

$\displaystyle \sum_{n\geq1} B_{2n+1} = 0 \qquad \textit{(naught series)}$

$\displaystyle \sum_{n\geq1} n\,B_{n+1} = -2\Psi^{(1)}(2) - \Psi^{(2)}(2) \\ = 2(\zeta(3)-\zeta(2)) = 2\zeta(3) - \dfrac{\pi^2}{3} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} n\,B_{2n} = \sum_{n\geq1} n\,B_{2n} = \zeta(3) - \dfrac{\zeta(2)}{2} + \dfrac{\zeta(0)}{2} \\ = \zeta(3) - \dfrac{\pi^2}{12} - \dfrac{1}{4} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} 2n\,B_{2n} = \sum_{n\geq1} 2n\,B_{2n} = 2\,\zeta(3) - \zeta(2) + \zeta(0) \\ = 2\,\zeta(3) - \dfrac{\pi^2}{6} - \dfrac{1}{2} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} (2n\!+\!1)\,B_{2n} = 2(\zeta(3)+\zeta(0)) = 2\,\zeta(3) - 1 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq2} n\,B_n = \sum_{n\geq1} (n\!+\!1)\,B_{n+1} = \dfrac{1}{2} - \Psi^{(1)}(2) - \Psi^{(2)}(2) \\ = 2\zeta(3) - \zeta(2) + \zeta(0) = 2\,\zeta(3) - \dfrac{\pi^2}{6} - \dfrac{1}{2} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq2} \dfrac{B_n}{n} = \sum_{n\geq1} \dfrac{B_{n+1}}{n\!+\!1} = \sum_{n\geq1} \dfrac{B_{2n}}{2n} \\ = -\Psi^{(0)}(1) + \zeta(0) = \zeta(1) + \zeta(0) = \gamma - \dfrac{1}{2} \quad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{2n}}{n} = -2\,\Psi^{(0)}(1) +2\,\zeta(0) = 2\,\zeta(1) - 1 = 2\gamma - 1 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq2} \dfrac{B_n}{n(n\!-\!1)} = \sum_{n\geq1} \dfrac{B_{n+1}}{n(n\!+\!1)} = -\dfrac{\ln(2\pi)}{2} \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq0} \dfrac{B_{2n}}{(2n)!} = \dfrac{1}{2} \coth{\dfrac{1}{2}} = \dfrac{1}{e\!-\!1} \,+\, \dfrac{1}{2} = \dfrac{1}{2} \, \dfrac{e\!+\!1}{e\!-\!1} \qquad \textit{(convergent)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{2n}}{(2n)!} = \dfrac{1}{2} \coth{\dfrac{1}{2}} \,-\, 1 = \dfrac{1}{e\!-\!1} \,-\, \dfrac{1}{2} = \dfrac{1}{2} \, \dfrac{3\!-\!e}{e\!-\!1} \quad \textit{(convergent)}$

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