Alternate Series of Bernoulli Numbers

SPECIES

$\displaystyle \sum_{n\geq0} (-1)^n \, B_n^{+} = -\psi_1(-1) = \zeta(2,1) - 1 = \zeta(2) - 1 = \dfrac{\pi^2}{6} - 1$ (stable)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, B_n^{+} = 1 + \psi_1(-1) = 2 - \zeta(2,1) = 2 - \zeta(2) = 2 - \dfrac{\pi^2}{6}$ (stable)

$\displaystyle \sum_{n\geq0} |B_{2n}| = 1 \,+\, \sum_{n\geq1} |B_{2n}| = 1 \,+\, \sum_{n\geq1} (-1)^{n-1} \, B_{2n} = 2 \,+\, i \left(\dfrac{1}{2} + \psi^{(1)}(-i)\right) \\ = 2 \,+\, i \left(\dfrac{1}{2} - \pi^2 \text{csch}^2\pi - \psi^{(1)}(1+i)\right)$ (stable)

Erratum: above sum corrected by Denis Rogov on 8 Nov 2023

$\displaystyle \sum_{n\geq1} |B_{2n}| = \sum_{n\geq1} (-1)^{n-1} \, B_{2n} = 1 \,+\, i \left(\dfrac{1}{2} + \psi^{(1)}(-i)\right) \\ = 1 \,+\, i \left(\dfrac{1}{2} - \pi^2 \text{csch}^2\pi - \psi^{(1)}(1+i)\right)$ (stable)