English Title

On the sum of series involving the Bernoulli numbers

Authors

Leonhard Euler

Enestrom Number

393

Fuss Index

183

Original Language

Latin

Published as

Journal article

Published Date

1770

Written Date

1768

Content Summary

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.

Record Created

2018-09-25

Included in

Mathematics Commons

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