I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers:
$$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 \right) ^{k}}{ \left( 2\,i-2\,k+1 \right) !}}{i}^{-1}=1/2\,\sum _{k=1}^{i}{\frac {{2}^{2\,k}B_{2k} }{ \left( 2\,k \right) !\, \left( 2\,i-2\,k+1 \right) !}}{i}^{-1}=\frac{1}{(2i+1)!}$$
I'm pretty sure that this formula is already known, but can't find any reference for it. Does somebody have any reference for it?
Thank you very much for any hint.