Dear Allan,
First of all, thank you for your 50 years of commitment and hard work
on your excellent Puzzle Corner.
You and I overlapped a little at MIT, but I was going for my PhD in
EE. I entered MIT as a PhD student in 1966.
I am writing to you about problem N/D 3. I had always been interested
in the various means, so this problem piqued by interest. So, I sent
it to a couple of colleagues, one of whom is Professor Amro El-Jaroudi
at the University of Pittsburgh.
Amro discovered two things. First, there is an error in the published
proof. (In the abbreviated form in TR, p. 63, right column, the
equation after "from which determine" is incorrect.) I sent Amro the
full proof on your web site, so he proceeded to write a correct proof
(attached).
The second thing that Amro discovered has to do with the limit of the
iteration for very large b0. The problem asks: "what is the limit
when a0=1 and b0 is large"? Amro and I both took this to mean that
the problem was seeking a formula for that limit when b0 is very
large, i.e., as it approached infinity. So, when I saw the solution
to the problem in the recent issue (and the expanded version on your
web site), I did not see any such formula. As it turns out, Amro had
already found that formula but did not submit it. He found that the
limit as b0 approached infinity is:
L(1, b0) => pi/2 * b0 / ln(b0)
For lower values of b0, the above formula appears to be always an
overestimate. Amro would be happy to send you a proof for that
formula if you wish.
Thanks again and all the best to you.
John