Let’s define partial sums of the harmonic series as follows:
,
where are positive integers. Here are a couple of facts that I didn’t know before reading Gamma (pages 24-25):
is never equal to an integer.
The only values of for which is an integer are 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.
I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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is never equal to an integer.
for which
is an integer are
and
.
Mean Green Math 
One thought on “What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 2”