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Servois' 1817 "Memoir on Quadratures" – Bernoulli Numbers

Author(s): 
Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University)

 

The Bernoulli Numbers are named in honor of Jakob Bernoulli (1655–1705), the older brother of Johann Bernoulli (1667–1748) and uncle of Daniel Bernoulli (1700–1782). He studied the formulas for the sums of powers and presented some results about them in his posthumous book Ars conjectandi [Bernoulli 1713]. Bernoulli was not the first person to study these formulas, but he did collect together all the formulas for the powers \(n=1, 2, 3, \ldots, 10\) and noticed a remarkable pattern in the coefficients of these formulas.

Figure 4: Portrait of Jakob Bernoulli. Public domain

Let's use modern notation and define

\[S_m(n) = \sum_{k=1}^n k^m = 1^m + 2^m + 3^m + \cdots + n^m.\]

Students usually learn the first few of these formulas in a calculus class and maybe prove them using mathematical induction:

\[\begin{array}{rll}S_1(n) &= {\displaystyle \frac{n(n+1)}{2}} &= \frac{1}{2} n^2 + \frac{1}{2} n,\\ S_2(n) &= {\displaystyle \frac{n(n+1)(2n+1)}{6}} &= \frac{1}{3} n^3 + \frac{1}{2} n^2 + \frac{1}{6} n,\\ S_3(n) &= {\displaystyle \left(\frac{n(n+1)}{2}\right)^2} & = \frac{1}{4} n^4 + \frac{1}{2} n^3 + \frac{1}{3} n^2.\end{array}\]

There are some patterns here. For example, the first term is always \(\frac{1}{m+1}n^{m+1}\) and the second is always \(\frac{1}{2}n^m\). With the help of the binomial coefficients

\[\binom{n}{k} = \frac{n\cdot(n-1) \cdots (n-k+1)}{k!}\]

Bernoulli eventually found that that there is a sequence of constants\(b_0, b_1, b_2, \ldots\), such that

\[ S_m(n) = \frac{1}{m+1} \sum_{k=0}^m b_k \binom{m+1}{k}  n^{m+1-k}.\]

These numbers are called the Bernoulli Numbers. They are usually denoted \(B_k\) instead of \(b_k\), but because Servois used the notation \(B_n\) for something slightly different, we are using \(b_k\) to denote the Bernoulli numbers as they are usually defined today.

It's easy to see from our knowledge of \(S_1(n)\), \(S_2(n)\) and \(S_3(n)\) that

\[b_0 = 1, \quad b_1 = \frac{1}{2}, \quad \mbox{and } b_2 = \frac{1}{6}.\]

What is much less obvious is that \(b_k\) is zero for odd values of \(k\) bigger than one. The even values are

\[b_2 = \frac{1}{6}, \quad b_4 = -\frac{1}{30}, \quad b_6 = \frac{1}{42}, \quad b_8 = -\frac{1}{30} \ldots.\]

There is an alternating sign, but otherwise the pattern followed by the Bernoulli numbers of even index is not a simple one. The Bernoulli numbers can be given by a simple recursion and they can also be found in a variety of infinite series. For example,

\[ \cot x = \frac{1}{x} \sum^{\infty}_{n=0}\frac{(-1)^n b_{2n} (2x)^{2n}}{(2n)!}.\,\,\,\,\,\,\,\,\,\,\mbox{(V)}\]

Because all the Bernoulli numbers of odd index are zero, except the first, some authors use \(B_n\) to denote \(b_{2n}\). When Servois introduced the Bernoulli numbers in formula (6) of his “Memoir on Quadratures” he did something similar; he made the definition \(B_n = |b_{2n}|\). With this notation, the MacLaurin series for cotangent is

\[\cot x = \frac{1}{x} \left[ 1 - \sum^{\infty}_{n=1}\frac{B_n (2x)^{2n}}{(2n)!} \right].\,\,\,\,\,\,\,\,\,\,\,\mbox{(VI)}\]

 

Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University), "Servois' 1817 "Memoir on Quadratures" – Bernoulli Numbers," Convergence (May 2019)

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