Bernoulli number

Bernoulli numbers B±
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

In mathematics, the Bernoulli numbers Bn 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.

The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by and ; they differ only for n = 1, where and . For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials , with and .[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[2][3][4] 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 for generating Bernoulli numbers with Babbage's machine;[5] it is disputed whether Lovelace or Babbage developed the algorithm. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

  1. ^ Cite error: The named reference Weisstein2016 was invoked but never defined (see the help page).
  2. ^ Cite error: The named reference Selin1997_891 was invoked but never defined (see the help page).
  3. ^ Cite error: The named reference SmithMikami1914_108 was invoked but never defined (see the help page).
  4. ^ Cite error: The named reference Kitagawa was invoked but never defined (see the help page).
  5. ^ Cite error: The named reference Menabrea1842_noteG was invoked but never defined (see the help page).