Monotonic Series of Bernoulli Numbers

$B_n$ denotes the n-th Bernoulli number.
$B_n^{+}$ denotes the n-th Bernoulli number in the context of $B_1^{+} = \frac{1}{2}$ .
$B_n^{-}$ denotes the n-th Bernoulli number in the context of $B_1^{-} = -\frac{1}{2}$ .
$E_n$ denotes the n-th Euler number.

$\zeta(x)$ denotes the Riemann zeta function.
$\zeta(x,a)$ denotes the Hurwitz zeta function.
$\Psi^{(0)}(x) \,=\, \frac{d}{dx} \ln\Gamma(x)$ denotes the digamma function.
$\Psi^{(1)}(x) \,=\, \frac{d}{dx}\Psi^{(0)}(x) \,=\, \frac{d^2}{dx^2} \ln\Gamma(x) \,=\, \zeta(2,x)$ denotes the trigamma function.

$\gamma = \zeta(1) \approx 0.577215664901532860606512090082402431\ldots$ denotes the Euler-Mascheroni constant.

SECTIONS

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

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

$\displaystyle \sum_{n\geq1} \dfrac{B_{2n}}{(2n)!} \, a^{2n} = \dfrac{a}{2} \, \dfrac{e^a+1}{e^a-1} - 1 = \dfrac{a}{2} \coth{\dfrac{a}{2}} - 1 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{2n}}{(2n)!} \, a^{2n-1} = \dfrac{1}{2} \, \dfrac{e^a+1}{e^a-1} - \dfrac{1}{a} = \dfrac{1}{2} \, \coth{\dfrac{a}{2}} - \dfrac{1}{a} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{2n}}{(2n)!} \, (\ln a)^{2n-1} = \dfrac{1}{2} \, \dfrac{a+1}{a-1} - \dfrac{1}{\ln a} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq0} \dfrac{B_{2n}}{(2n+1)!} \, a^{2n} = \dfrac{1}{2a} \, \coth{\dfrac{a}{2}} - \left(\dfrac{1}{2} \, \text{csch} \dfrac{a}{2}\right)^2 \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{2n}}{(2n-1)!} \, a^{2n-1} = \dfrac{1}{2} \, \coth{\dfrac{a}{2}} - \dfrac{a}{4} \, \text{csch}^2 \dfrac{a}{2} \qquad \textit{(stable)}$

$\displaystyle \sum_{n\geq1} n\,B_{n+1} \, a^n = -\dfrac{2}{a^2}\,\Psi^{(1)}\!\left(1\!+\!\dfrac{1}{a}\right) - \dfrac{1}{a^3}\,\Psi^{(2)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} n\,B_{n+1} \, a^{n+1} = -\dfrac{2}{a}\,\Psi^{(1)}\!\left(1\!+\!\dfrac{1}{a}\right) - \dfrac{1}{a^2}\,\Psi^{(2)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} B_{n+1} \, a^n = \sum_{n\geq2} B_n \, a^{n-1} = \dfrac{1}{2} + \dfrac{1}{a^2}\,\Psi^{(1)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} B_{n+1} \, a^{n+1} = \sum_{n\geq2} B_n \, a^n = \dfrac{a}{2} + \dfrac{1}{a}\,\Psi^{(1)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{n+1}}{n\!+\!1} \, a^{n-1} = \dfrac{1}{2a} - \dfrac{1}{a^2}\,\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{n+1}}{n\!+\!1} \, a^n = \dfrac{1}{2} - \dfrac{1}{a}\,\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{n+1}}{n\!+\!1} \, a^{n+1} = \sum_{n\geq2} \dfrac{B_n}{n} \, a^n = \dfrac{1}{2} - \dfrac{1}{a}\,\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{n+1}}{n(n\!+\!1)} \, a^{n-1} = \dfrac{\ln a}{2a} - \dfrac{\ln(2\pi)}{2a} + \dfrac{1}{a}\,\ln\Gamma\!\left(1\!+\!\dfrac{1}{a}\right) \quad \textit{(unstable)}$

$\displaystyle \sum_{n\geq1} \dfrac{B_{n+1}}{n(n\!+\!1)} \, a^n = \dfrac{\ln a}{2} - \dfrac{\ln(2\pi)}{2} + \ln\Gamma\!\left(1\!+\!\dfrac{1}{a}\right) \qquad \textit{(unstable)}$

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