A summation formula for convergence acceleration of some Dirichlet and related series

Our purpose is to prove a very simple but, however, very general transformation formula which can easily and very generally be applied to Dirichlet and related series with the aim to accelerate their convergence. In particular, the formula will be used to get series expressions for some famous constants as, for instance, log2 and pi, as well as the Euler and Catalan constants and zeta(2), zeta(3), zeta(5), zeta(1/2), zeta(-1/2), zeta ''(2), zeta'(2) and zeta'(1/2). We will see that the formula can also be used to derive analogous formulae for other Dirichlet and related series as well; especially, a number of estimates for the logarithmic derivative of Euler's Gamma function will turn out as a further implication.