# Bernoulli numbers

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The sequence of rational numbers discovered by Jacob Bernoulli  in connection with the calculation of the sum of equal powers of natural numbers:  The values of the first Bernoulli numbers are:   All odd-indexed Bernoulli numbers except for are zero, and the signs of alternate. Bernoulli numbers are the values of the Bernoulli polynomials at : ; they also often serve as the coefficients of the expansions of certain elementary functions into power series. Thus, for example, (the so-called generating function of the Bernoulli numbers);  L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values of the Riemann zeta-function for even : Bernoulli numbers are used to express many improper integrals, such as Certain relationships involving Bernoulli numbers are: (the recurrence formula);  The estimates: hold. Extensive tables of Bernoulli numbers are available; for instance,  contains accurate values of for and approximate values for .

Bernoulli numbers have found many applications in mathematical analysis, number theory and approximate calculations.

How to Cite This Entry:
Bernoulli numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_numbers&oldid=15872
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article