A succinct method for investigating the sums of infinite series through differential formulae

Translation of "Methodus succincta summas serierum infinitarum per formulas differentiales investigandi" (1780). Euler wants to represent some given series of functions S(x)=X(x)+X(x+1)+X(x+2)+etc. in a different way. He writes S as a series in derivatives of X with unknown coefficients. He makes a generating function V(z) out of these coefficients, which is the same as a generating function that involves the Bernoulli numbers.