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

TL;DR

This paper explores a concise method, based on differential formulae, for analyzing infinite series sums, extending Euler's approach with generating functions involving derivatives and Bernoulli numbers.

Contribution

It introduces a novel succinct technique for investigating infinite series sums using differential formulae and generating functions, building on Euler's classical methods.

Findings

Derived a new differential formula-based method for series summation.

Connected the method with Bernoulli numbers through generating functions.

Demonstrated the approach's effectiveness in representing series sums.

Abstract

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.