Bernoulli number

The Bernoulli numbers B_n 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 b_n is used in order to distinguish these from the Bell numbers.

The first eleven Bernoulli numbers are listed below.

n	Bn
0	1
1	-1/2
2	1/6
3	0
4	-1/30
5	0
6	1/42
7	0
8	-1/30
9	0
10	5/66

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 B_n = 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.