# History of Bernoulli numbers

I have been trying to understand what is the meaning of Bernoulli numbers, but to my mind it has been obscured behind complicated formulas without much explaination. I presume finding the history could be of use in understanding.

The formulas I found are: $$\frac{x}{1-e^{-x}} = \frac{x}{2}(\coth{\frac{x}{2}} + 1) = \sum^\infty_{k=0}{\frac{B_k x^k}{k!}}$$

One answer in math stack exchange said:

It is a analytical continuation of the sum of the infinite series $$\frac{1}{n^s}$$, where the real part of the sum converges.

Is this true? I mean, $$\frac{x}{1-e^x}$$ does resemble $$\frac{1}{1-x}$$, but where do Bernoulli numbers come in here?

• @Mauricio Yup, when I wrote it I missed the $\frac{x}{2}$ factor, now I edited it in, thanks! Jan 22 at 8:38
• What's the question here? If you're asking about the MSE answer, then your question belongs to MSE rather than HSMSE (and my answer would be: the linked answer is weird and misleading at best). Jan 22 at 9:53
• If you're asking for references on the history of Bernoulli numbers, you can just start with the Wikipedia page you linked to. There's a section "history" with a number of sources. I can also recommend a nice essay by James Propp. Jan 22 at 9:55
• @MichałMiśkiewicz Alright, sorry for the ambiguity of the question, I guess what I wanted was the easiest problem solved, which used Bernoulli numbers, I assumed it to be the problem which historically introduced them. The wikipedia and essay were useful! After some thought I now understand that the Bernoulli numbers are the coefficients for a sum of natural numbers to a power formula. Now I have a question on how he calculated the Bernoulli numbers, where those formulas for calculating them came from, but I suppose that this is a question for math stack exchange. Jan 22 at 13:16
• @Gustamons As Wikipedia says, the formulas appear in the posthumously published Ars Conjectandi (1713), dealing with permutations and combinations. Specifically in chapter 3 "De Combinationibus secundum singulos exponentes seorsim; ubi de Numeris Figuratis eorumque proprietatibus agitur" (Of the Combinations according to the individual exponents separately; where Figured Numbers and their properties are discussed), pp. 86-99. The Summae Potestatum are on p. 97. Jan 22 at 22:05