Bernoulli number

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 values of the Riemann zeta function.

Bernoulli numbers $B \pm n$
$n$	fraction	decimal
0	1	+1.000000000
1	±1/2	±0.500000000
2	1/6	+0.166666666
3	0	+0.000000000
4	−1/30	−0.033333333
5	0	+0.000000000
6	1/42	+0.023809523
7	0	+0.000000000
8	−1/30	−0.033333333
9	0	+0.000000000
10	5/66	+0.075757575
11	0	+0.000000000
12	−691/2730	−0.253113553
13	0	+0.000000000
14	7/6	+1.166666666
15	0	+0.000000000
16	−3617/510	−7.092156862
17	0	+0.000000000
18	43867/798	+54.97117794
19	0	+0.000000000
20	−174611/330	−529.1242424

The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by $B_{n}^{-{}}$ and $B_{n}^{+{}}$ ; they differ only for $n = 1$ , where $B_{1}^{-{}}=-1/2$ and $B_{1}^{+{}}=+1/2$ . For every odd $n > 1$ , $B n = 0$ . For every even $n > 0$ , $B n$ is negative if $n$ is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials $B_{n}(x)$ , with $B_{n}^{-{}}=B_{n}(0)$ and $B_{n}^{+}=B_{n}(1)$ .

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm prepared by Babbage for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

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