Bernoulli numbers

Bernoulli numbers are a sequence of rational numbers defined by a generating function and used in formulas for sums of powers, series, and special values of zeta functions.

Bernoulli generating function: The power series identity that defines Bernoulli numbers through the expansion of x/(e^x−1). Its coefficients are the Bernoulli numbers, making this the sequence’s defining construction.
Jakob Bernoulli: A Swiss mathematician whose posthumously published work introduced the systematic study of sums of powers. His Ars Conjectandi presented the numbers now bearing his surname.
Faulhaber polynomials: The polynomials in n that give sums of the first n positive integers raised to a fixed power. Their coefficients encode the Bernoulli numbers in explicit power-sum formulas.
Bernoulli number convention: Conventions for indexing Bernoulli numbers and assigning the value of B₁, which may be 1/2 or −1/2. The two common B₁ conventions differ, while the higher numbers agree.
Bernoulli recurrence: A recurrence expressing each Bernoulli number using earlier terms and binomial coefficients. It provides a direct way to calculate the sequence without extracting power-series coefficients.
Ars Conjectandi: A 1713 mathematical work by Jakob Bernoulli on probability, combinatorics, and sums of powers. Its discussion of power sums made Bernoulli numbers widely known.
Euler–Maclaurin summation: A method for approximating a sum by an integral plus endpoint corrections and a controlled remainder. Bernoulli numbers determine the coefficients of its successive endpoint corrections.
Euler numbers: An integer sequence defined by the Taylor expansion of the secant and hyperbolic secant functions. They resemble Bernoulli numbers in generating-function methods but govern different expansions.
Faulhaber's formula: A formula expressing the sum of the first n positive integers raised to a fixed power as a polynomial in n. Bernoulli numbers supply the coefficients of these power-sum polynomials.
Guillaume de l'Hôpital: A French mathematician who published the first textbook on differential calculus in 1696. His textbook included an early account of a power-sum formula later associated with Bernoulli numbers.

Linked from 13 pages