Board index Computation of Series Series involving harmonic number and hyperbolic sine functi

Series involving harmonic number and hyperbolic sine functi

Post your questions related to Computation of Series here.

Moderators: galactus, sos440, zaidalyafey



Posts: 105
I deduce the formula that
$$\sum\limits_{n = 1}^\infty {\frac{{\pi {H_n}}}{{n\sinh \left( {n\pi } \right)}}} = \frac{1}{2}\zeta \left( 3 \right) + \pi \sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}\sinh \left( {n\pi } \right)}}} + \frac{{{\pi ^2}}}{2}\sum\limits_{n = 1}^\infty {\frac{{\cosh \left( {n\pi } \right)}}{{n{{\sinh }^2}\left( {n\pi } \right)}}} + \sum\limits_{n,k = 1}^\infty {\frac{{k\pi \coth \left( {k\pi } \right)}}{{n\left( {{n^2} + {k^2}} \right)}}{{\left( { - 1} \right)}^k}}$$
However, the sum on the right hand side
$$\sum\limits_{n,k = 1}^\infty {\frac{{k\pi \coth \left( {k\pi } \right)}}{{n\left( {{n^2} + {k^2}} \right)}}{{\left( { - 1} \right)}^k}}=? $$
have further simplifications?

Return to Computation of Series

cron