Rational Zeta Series - Elementary Series

Elementary Series

For integer m, one has

For m=2, a number of interesting numbers have a simple expression as rational zeta series:

and

where γ is the Euler–Mascheroni constant. The series

follows by summing the Gauss–Kuzmin distribution. There are also series for π:

and

being notable because of its fast convergence. This last series follows from the general identity

\sum_{n=1}^\infty (-1)^{n} t^{2n} \left = \frac{t^2}{1+t^2} + \frac{1-\pi t}{2} - \frac {\pi t}{e^{2\pi t} -1}

which in turn follows from the generating function for the Bernoulli numbers

Adamchik and Srivastava give a similar series

\sum_{n=1}^\infty \frac{t^{2n}}{n} \zeta(2n) = \log \left(\frac{\pi t} {\sin (\pi t)}\right)

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