Editing Bernoulli number (section)

=== Reconstruction of "Summae Potestatum" ===
[[File:JakobBernoulliSummaePotestatum.png|thumb|right|upright=1.5|Jakob Bernoulli's "Summae Potestatum", 1713{{efn|Translation of the text: 
" ... And if [one were] to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list:<br>
Sums of powers<br>
<math>\textstyle \int n = \sum_{k=1}^n k = \frac {1}{2} n^2 + \frac {1}{2} n </math><br>
::::⋮
<math>\textstyle \int n^{10} = \sum_{k=1}^n k^{10} = \frac {1}{11} n^{11} + \frac {1}{2} n^{10} + \frac {5}{6} n^9 - 1 n^7 + 1 n^5 - \frac {1}{2} n^3 + \frac {5}{66} n </math><br>
Indeed [if] one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these circuitous computations:  For [if] <math>\textstyle c </math> is taken as the exponent of any power, the sum of all <math>\textstyle n^c </math> is produced or<br>
<math>\textstyle \int n^c = \sum_{k=1}^n k^c = \frac {1}{c+1} n^{c+1} + \frac {1}{2} n^c + \frac {c}{2} An^{c-1} + \frac {c(c-1)(c-2)}{2\cdot 3\cdot4} Bn^{c-3} + \frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4 \cdot 5 \cdot 6} Cn^{c-5} + \frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4 \cdot 5 \cdot 6 \cdot 7  \cdot 8} Dn^{c-7} + \cdots </math><br>
and so forth, the exponent of its power <math>n</math> continually diminishing by 2 until it arrives at <math>n</math> or <math>n^2</math>.  The capital letters <math>\textstyle  A, B, C, D, </math> etc. denote in order the coefficients of the last terms for <math>\textstyle \int n^2 , \int n^4 , \int n^6 , \int n^8 </math>, etc. namely<br>
<math>\textstyle  A = \frac {1}{6} , B = - \frac {1}{30} , C = \frac {1}{42} , D = - \frac {1}{30} </math>."<br>
[Note:  The text of the illustration contains some typos:  ''ensperexit'' should read ''inspexerit'', ''ambabimus'' should read ''ambagibus'', ''quosque'' should read ''quousque'', and in Bernoulli's original text ''Sumtâ'' should read ''Sumptâ'' or ''Sumptam''.]
* {{citation |last=Smith |first=David Eugene |date= 1929 |chapter=Jacques (I) Bernoulli: On the 'Bernoulli Numbers' |title= A Source Book in Mathematics |chapter-url=https://archive.org/details/sourcebookinmath00smit/page/85 |location=New York |publisher=McGraw-Hill Book Co. |pages=85–90 }}
* {{citation |last=Bernoulli |first=Jacob |date=1713 |title=Ars Conjectandi |url=https://archive.org/details/jacobibernoulli00bern/page/97 |location=Basel |publisher=Impensis Thurnisiorum, Fratrum |pages=97–98 |language=la |doi=10.5479/sil.262971.39088000323931}}
}}]]

The Bernoulli numbers {{OEIS2C|id=A164555}}(n)/{{OEIS2C|id=A027642}}(n) were introduced by Jakob Bernoulli in the book ''[[Ars Conjectandi]]'' published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted {{math|''A''}}, {{math|''B''}}, {{math|''C''}} and {{math|''D''}} by Bernoulli are mapped to the notation which is now prevalent as {{math|''A'' {{=}} ''B''<sub>2</sub>}}, {{math|''B'' {{=}} ''B''<sub>4</sub>}}, {{math|''C'' {{=}} ''B''<sub>6</sub>}}, {{math|''D'' {{=}} ''B''<sub>8</sub>}}. The expression {{math|''c''·''c''−1·''c''−2·''c''−3}} means {{math|''c''·(''c''−1)·(''c''−2)·(''c''−3)}} – the small dots are used as grouping symbols. Using today's terminology these expressions are [[Pochhammer symbol|falling factorial powers]] {{math|''c''<sup>{{underline|''k''}}</sup>}}. The factorial notation {{math|''k''!}} as a shortcut for {{math|1 × 2 × ... × ''k''}} was not introduced until 100 years later. The integral symbol on the left hand side goes back to [[Gottfried Wilhelm Leibniz]] in 1675 who used it as a long letter {{math|''S''}} for "summa" (sum).{{efn|The {{harvp|''Mathematics Genealogy Project''|n.d.}} shows Leibniz as the academic<!--not doctoral--> advisor of Jakob Bernoulli. See also {{harvp|Miller|2017}}.}} The letter {{math|''n''}} on the left hand side is not an index of [[summation]] but gives the upper limit of the range of summation which is to be understood as {{math|1, 2, ..., ''n''}}. Putting things together, for positive {{math|''c''}}, today a mathematician is likely to write Bernoulli's formula as:

: <math> \sum_{k=1}^n k^c = \frac{n^{c+1}}{c+1}+\frac 1 2 n^c+\sum_{k=2}^c \frac{B_k}{k!} c^{\underline{k-1}}n^{c-k+1}.</math>

This formula suggests setting {{math|''B''<sub>1</sub> {{=}} {{sfrac|1|2}}}} when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the [[falling factorial#Real_numbers_and_negative_n|falling factorial]] {{math|''c''<sup>{{underline|''k''−1}}</sup>}} has for {{math|''k'' {{=}} 0}} the value {{math|{{sfrac|1|''c'' + 1}}}}.{{sfnp|Graham|Knuth|Patashnik|1989|loc=Section 2.51}} Thus Bernoulli's formula can be written
: <math> \sum_{k=1}^n k^c = \sum_{k=0}^c \frac{B_k}{k!}c^{\underline{k-1}} n^{c-k+1}</math>

if {{math|''B''<sub>1</sub> {{=}} 1/2}}, recapturing the value Bernoulli gave to the coefficient at that position.

The formula for <math>\textstyle \sum_{k=1}^n k^9</math> in the first half of the quotation by Bernoulli above contains an error at the last term; it should be <math>-\tfrac {3}{20}n^2</math> instead of <math>-\tfrac {1}{12}n^2</math>.