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In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
The Bernoulli numbers have various definitions (see Bernoulli number#Definitions), such as that they are the coefficients of the exponential generating function = ( +) = =!. Then Faulhaber's formula is that ∑ k = 1 n k p = 1 p + 1 ∑ k = 0 p ( p + 1 k ) B k n p − k + 1 . {\displaystyle \sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{k=0}^{p ...
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt and Thomas Clausen . Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B 2n for every prime p such that p − 1 divides 2n, then we obtain an integer; that is,
On a note more distantly related to combinatorics, the second section also discusses the general formula for sums of integer powers; the free coefficients of this formula are therefore called the Bernoulli numbers, which influenced Abraham de Moivre's work later, [16] and which have proven to have numerous applications in number theory. [22]
Bernoulli numbers can be calculated in many ways, but Lovelace deliberately chose an elaborate method in order to demonstrate the power of the engine. In Note G, she states: "We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we ...
Education: University of Basel (D.Th., 1676; Dr. phil. hab., 1684) Known for: Bernoulli differential equation Bernoulli numbers Bernoulli's formula Bernoulli polynomials
Gregory coefficients G n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, [1] [2] [3] [4] [5 ...
Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. [ 4 ] [ 5 ] Bernoulli's principle can be derived from the principle of conservation of energy .