Bernoulli Numbers

Abstract

In his posthumous book Ars Conjectandi [16] published in 1713 (the law of large numbers in probability theory is stated in this book), Jakob Bernoulli introduced the Bernoulli numbers in connection to the study of the sums of powers of consecutive integers $1^{k} + 2^{k} + \cdots + n^{k}$. After listing the formulas for the sums of powers

$$\displaystyle{\sum _{i=1}^{n}i = \frac{n(n + 1)} {2},\ \ \sum _{i=1}^{n}i^{2} = \frac{n(n + 1)(2n + 1)} {6},\ \ \sum _{i=1}^{n}i^{3} = \left (\frac{n(n + 1)} {2} \right )^{2},\ldots }$$

up to k = 10 (Bernoulli expresses the right-hand side without factoring), he gives a general formula involving the numbers which are known today as Bernoulli numbers. Bernoulli then explains how these numbers are determined inductively, and emphasizes how his formula ((1.1) below) is useful for computing the sum of powers. He claims that he did not take “a half of a quarter of an hour” to compute the sum of tenth powers of 1 to 1, 000, which he computed correctly as 91409924241424243424241924242500.