Bernoulli number

Etymology
Named after Swiss mathematician (1654–1705), who discovered the numbers independently of and at about the same time as Japanese mathematician.

The numbers first appeared as coefficients in Bernoulli's formulae for the sum of the first n positive integers, each raised to a given power. The sequence was subsequently found in numerous other contexts.

Noun

 * 1)  Any one of the rational numbers in a sequence of such that appears in numerous contexts, including formulae for sums of integer powers and certain power series expansions.
 * 2) * 1993,, Complex Analysis, Springer, 3rd Edition, page 418,
 * The assertion about the value of the zeta function at negative integers then comes immediately from the definition of the Bernoulli numbers in terms of the coefficients of a power series, namely
 * $$\frac t {e^t - 1} = \sum_{n=0}^\infty B_n\frac{t^n}{n!}$$

Usage notes

 * For odd values of $$n$$ greater than 1, $$B_n=0,$$ and many formulae involve only "even-index" Bernoulli numbers. Consequently, some authors ignore these values and write $$B_n$$ to mean what, properly speaking, is $$B_{2n}$$.
 * A sign convention affects the value assigned for $$n=1$$. The modern convention is that $$\textstyle B_1=-\frac {1} 2$$. An older convention, used by  and some older textbooks, has that $$\textstyle B_1^{*}=+\frac {1} 2$$.
 * The modified symbol $$\textstyle B^{*}$$ indicates the older convention is being used.
 * Alternatively, the notations $$B^{-}$$ and $$B^{+}$$ can be used (where $$\textstyle B_1^{-}=-\frac {1} 2$$ and $$\textstyle B_1^{+}=+\frac {1} 2$$).
 * The Bernoulli numbers may be regarded as special values of the Bernoulli polynomials $$B_n(x)$$, with $$B_n=B_n(0)$$ and $$B^{*}_n=B^{*}_n(0)$$.
 * Note that the notations for Bernoulli numbers and Bernoulli polynomials are very similar.
 * Note as well that the letter $$B$$ is used also for s and.
 * Places where Bernoulli numbers appear include:
 * Bernoulli's formula for the sum of the mth powers of the first n positive integers (also called Faulhaber's formula, although Faulhaber did not explore the properties of the coefficients).
 * Taylor series expansions of the tangent and hyperbolic tangent functions.
 * Formulae for particular values of the Riemann zeta function.
 * The residual error of partial sums of certain power series:
 * In particular, consider the series $$\textstyle\sum_{n=1}^\infty \frac 1{n^2}=\frac{\pi^2}{6}$$. The partial sum $$\textstyle S_n = \sum_{k=1}^n\frac 1{k^2}$$ differs from the limit value by $$\textstyle E_n=\sum_{k=0}^\infty \frac{B_k}{n^k}$$.
 * The.