I got a note after my last post that reminded me how little Julian Havel seems to be known in the US. Denise, of Let's Play Math, related that her local library had NO books by him. Impossible... No, I don't mean that it is impossible they have no books by him (it should be)...but that is one of his really good books.
Havel is (or was) a headmaster at a British College (sort of like the last year of our HS in the US).
In his book, Impossible, he has a chapter on the Harmonic Series in which he mentions a couple of more interesting notes I might have added for students.
For example, it is quite easy to see that H1= 1, and H2= 3/2. H6= 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 2 9/20... and those three are the only ones in the entire infinite series that end in a terminating decimal. Havel states that, "it can be proved" but does not... anyone know an tractable proof for that to approach Pre-calc students?
Another quick point to mention is the extremely slow rate at which the function diverges.... Ralph Boas Jr was co-finder of the smallest n for which Hn> 100. It turns out it takes more than 15 1042 terms to reach 100. Which means it is adding less than .01 each step... and yet... as Borovik cautions us... It Diverges..