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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.

BORDER CELLSPACING=0 CELLPADDING=0 width="15%" BGCOLOR="#EEEEEE" >
n B_n

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.

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