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Bernoulli number

The Bernoulli numbers Bn are named after Jakob Bernoulli, because he was the first to study them. They were named this way by Abraham De Moivre. The Bernoulli numbers are defined by

<math>
\frac{x}{e^x-1} = \sum_{n=0}^{\infin} \frac{B_n x^n}{n!} </math> for all values of x of absolute value less than 2π (2π is the radius of convergence of this power series).

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

 
The first eleven Bernoulli numbers are listed below.

nBn
01
1-1/2
21/6
30
4-1/30
50
61/42
70
8-1/30
90
105/66

One can calculate the Bernoulli numbers using the following recursive formula.

<math>
B_n = -n! \sum^{n-1}_{k=0} \frac{1}{k!(n+1-k)!} B_k </math>

It turns out that Bn = 0 whenever n is odd and n ≥ 3.

The Bernoulli numbers appear in the Taylor series expansion of the tangent and hyperbolic tangent functions, in the Euler-Maclaurin formula, and in expressions of certain values of the Riemann zeta function.



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