Re: A question on Reimann (ref Prime Obsession by John Derbyshire)

In article
<88801f5a-262a-4149-a9e7-1eab2161e1ed@xxxxxxxxxxxxxxxxx
groups.com>,
drmwecker@xxxxxxxxx wrote:

Besides the more obvious definitional meaning given by another poster,
here is one that I found intriguing since I was an undergrad:

The p-series (series sum 1/n^p, n = 1... inf) is well-known to
converge
if and only if p > 1. For p = 1 we have the harmonic series,
divergent.
For p = 1 + e, with e > 0 tiny, we have convergence.

On the other hand, series sum 1/(n*ln(n)), n = 1... inf, diverges
(for instance, by the integral test).

This means that ln(n) becomes smaller than any power n^e.

For more on this, I recommend the MAA book "Real Infinite Series"
by Bonar and Khoury, which came out in the last two years or so.

Best, Mike

Dr. Michael W. Ecker
Associate Professor of Mathematics
Pennsylvania State University
Wilkes-Barre Campus
Lehman, PA 18627

Better e-mail address is DrMWEcker at aol dot com


He writes "[Natural] Log x increases slower than any power of x"

More specifically set a > 0 and define

u_n = n.log(n).log(log(n)).log(log(log(n)))...(log(log(log(...(log(n))...))))^a
v_n = n.log(n).log(log(n)).log(log(log(n)))...log(log(log(...(log(n))...)))

sum_n 1/u_n converges
sum_n 1/v_n divverges

sum_n 1/(n.log(n).(log(log(n)))^2) converges
and needs 10^10^86 terms for two digit accuracy.

--
Michael Press
.