On the sum of series involving the Bernoulli numbers
In Chapter 5 of his Institutiones calculi differentialis (E212), Euler shows for the first time how the sequences of coefficients that arise in ζ(2n), the Euler-Maclaurin formula, and the Taylor series expansions of certain trigonometric functions are all related to the coefficients that Jakob Bernoulli had discovered in his book on probability, Ars Conjectandi. There, Euler named them the "Bernoulli numbers" and showed how all these different coefficients are related. In this paper, Euler again explores these connections, showing how the Bernoulli numbers are related to ζ(2n) and how they arise in certain integrals. He also derives some recurrence relations for Bernoulli numbers.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 14, pp. 129-167.
Opera Omnia Citation
Series 1, Volume 15, pp.91-130.