When teaching students mathematical induction, the following series (well, at least the first two or three) are used as typical examples:
What I didn’t know (Gamma, page 81) is that Johann Faulhaber published the following cute result in 1631 (see also Wikipedia): If is odd, then
where is a polynomial. For example, to match the above examples, and . Furthermore, if is even, then
where again is a polynomial. For example, to match the above examples, and .
When I researching for my series of posts on conditional convergence, especially examples related to the constant , the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.
Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.
In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.