Home > Bernoulli number

In mathematics, the Bernoulli numbers B_n were first discovered in connection with the closed forms of the sums

for various fixed values of n. The closed forms are always polynomials in m of degree n+1 and are called Bernoulli polynomials. The coefficients of the Bernoulli polynomials are closely related to the Bernoulli numbers, as follows:

For example, taking n to be 1, we have 0 + 1 + 2 + ... + (m−1) = 1/2 (B₀ m² + 2 B₁ m¹) = 1/2 (m² − m).

The Bernoulli numbers were first studied by Jakob Bernoulli, after whom they were named by Abraham de Moivre.

Bernoulli numbers may be calculated by using the following recursive formula:

plus the initial condition that B₀ = 1.

The Bernoulli numbers may also be defined using the technique of generating functions. Their exponential generating function is x/(e^x − 1), so that:

for all values of x of absolute value less than 2π (the radius of convergence of this power series).

Sometimes the lower-case b_n is used in order to distinguish these from the Bell numbers.

The first few Bernoulli numbers (sequences A027641 and A027642 in OEIS) are listed below.

It can be shown that B_n = 0 for all odd n other than 1. The appearance of the peculiar value B₁₂ = −691/2730 signals that the values of the Bernoulli numbers have no elementary description; in fact they are essentially values of the Riemann zeta function at negative integers, and are associated to deep number theoretic properies, and so cannot be expected to have a trivial formulation.

The Bernoulli numbers also appear in the Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point expansion of the tangentIn mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. Geometry In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the poi and hyperbolic tangent functions, in the Euler-Maclaurin formulaIn mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the, and in expressions of certain values of the Riemann zeta function.

In note G of Ada Byron's notes on the analytical engineIn 1840 Charles Babbage was invited to give a seminar at the University of Turin about his analytical engine. Luis Menabrea, a young Italian engineer wrote up Babbage's lecture in French, and this transcript was subsequently published in the Bibliotheque from 1842Events February 21 John J. Greenough patents the sewing machine. March 5 Over 500 Mexican troops led by Rafael Vasquez invade Texas briefly occupy San Antonio and then head back to the Rio Grande. This is the first such invasion since the Texas Revolution an algorithmFlowcharts were often used to represent algorithms. An algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will result in a corresponding recognisable end-state (contrast with heuristic). Algor for computer generated Bernoulli numbers was described for the first time.