What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 8

I had always wondered how the constant \gamma can be computed to high precision. I probably should have known this already, but here’s one way that it can be computed (Gamma, page 89):

\gamma = \displaystyle \sum_{k=1}^n \frac{1}{k} - \ln n - \sum_{k=1}^{\infty} \frac{B_{2k}}{2k \cdot n^{2k}},

where B_{2k} is the 2kth Bernoulli number.


green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, 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.

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  1. What I Learned by Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Index | Mean Green Math

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