Bernoulli numbers and Euler’s contributions to infinite series

Abstract

This expository study focuses on the origin and mathematical development of the sequence of rational Bernoulli numbers. Johann Faulhaber developed foundational formulas for power sums of positive integers, which later enabled Leonhard Euler to systematically develop and apply Bernoulli numbers. This study presents Euler's use of Bernoulli numbers as a bridge between finite sums and infinite series, emphasizing their role in generating functions and in the evaluation of the Riemann zeta function at positive even integers. It further elucidates the relation between the zeta function and Bernoulli numbers and interprets how Leonhard Euler continued to expand the theory by deriving exponential generating functions, highlighting their significance in analytic number theory.