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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 ...
However, the principle can be applied to various types of flow within these bounds, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach ...
The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values. The Beta distribution is the conjugate prior of the Bernoulli distribution. [5] The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.
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,
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions , and with the Euler–MacLaurin formula .
Take a look at the 10 countries with the highest number of billionaires, plus a few names of the wealthy that you might recognize. 1. The United States. Art Wager/istockphoto.
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of +. It is often employed in real analysis . It has several useful variants: [ 1 ]