Editing Bernoulli number (section)

{{Short description|Rational number sequence}}
{{Use American English|date = March 2019}}
{{Use shortened footnotes|date=April 2021}}
{| class=wikitable style="text-align: right; float:right; clear:right; margin-left:1em;"
|+ Bernoulli numbers {{math|''B''{{su|p=±|b=''n''}}}}
|- 
! {{mvar|n}} !! fraction !! decimal
|- decimal
| 0 || 1 || +1.000000000 
|-
| 1 || ±{{sfrac|1|2}} || ±0.500000000
|-
| 2 || {{sfrac|1|6}} || +0.166666666
|- style="background:#ABE"
| 3 || 0 || +0.000000000
|-
| 4 || −{{sfrac|1|30}} || −0.033333333
|- style="background:#ABE"
| 5 || 0 || +0.000000000
|-
| 6 || {{sfrac|1|42}} || +0.023809523
|- style="background:#ABE"
| 7 || 0 || +0.000000000
|-
| 8 || −{{sfrac|1|30}} || −0.033333333
|- style="background:#ABE"
| 9 || 0 || +0.000000000
|-
| 10 || {{sfrac|5|66}} || +0.075757575
|- style="background:#ABE"
| 11 || 0 || +0.000000000
|-
| 12 || −{{sfrac|691|2730}} || −0.253113553
|- style="background:#ABE"
| 13 || 0 || +0.000000000
|-
| 14 || {{sfrac|7|6}} || +1.166666666
|- style="background:#ABE"
| 15 || 0 || +0.000000000
|-
| 16 || −{{sfrac|3617|510}} || −7.092156862
|- style="background:#ABE"
| 17 || 0 || +0.000000000
|-
| 18 || {{sfrac|43867|798}} || +54.97117794
|- style="background:#ABE"
| 19 || 0 || +0.000000000
|-
| 20 || −{{sfrac|174611|330}} || −529.1242424
|}

In [[mathematics]], the '''Bernoulli numbers''' {{math|''B''<sub>''n''</sub>}} are a [[sequence]] of [[rational number]]s which occur frequently in [[Mathematical analysis|analysis]]. The Bernoulli numbers appear in (and can be defined by) the [[Taylor series]] expansions of the [[tangent function|tangent]] and [[Hyperbolic function|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 <math>B^{-{}}_n</math> and <math>B^{+{}}_n</math>; they differ only for {{math|''n'' {{=}} 1}}, where <math>B^{-{}}_1=-1/2</math> and <math>B^{+{}}_1=+1/2</math>. For every odd {{math|''n'' > 1}}, {{math|''B''<sub>''n''</sub> {{=}} 0}}. For every even {{math|''n'' > 0}}, {{math|''B''<sub>''n''</sub>}} is negative if {{math|''n''}} is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the [[Bernoulli polynomials]] <math>B_n(x)</math>, with <math>B^{-{}}_n=B_n(0)</math> and <math>B^+_n=B_n(1)</math>.{{r|Weisstein2016}}

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{{r|Selin1997_891|SmithMikami1914_108|Kitagawa}} 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 [[Charles Babbage|Babbage]]'s machine;{{r|Menabrea1842_noteG}} it is disputed [[Ada Lovelace#Controversy over contribution|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]].