This is leading to the question: What is the most slowly converging series? --Gene ________________________________ From: victor miller <victorsmiller@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Fri, December 11, 2009 12:44:05 PM Subject: Re: [math-fun] Random Numbers I don't know about the false consequences, but, for example Persi Diaconis' Ph.D. thesis dealt with probability measures on the natural numbers (once one has a probability measure on the natural numbers, it's easy to turn it into one for all the integers). One can start with trying to define a set A of natural numbers as measurable if the limit lim_{n-->infinity} A(n)/n exists, where A(n) means the number of elements of A which are <=n. When the limit exists this is also the measure. Unfortunately, lots of interesting sets are not measurable -- e.g. the set of natural numbers whose first decimal digit is 1. However, one can enlarge the collection of measurable sets A to those where the following limit exists: lim_{n-->infinity} sum_{k in A, k<=n} 1/k/sum_{k <=n} 1/k One can show that any set measurable the first way ("natural density") is measurable in the second ("harmonic density") and has the same measure. The first digit set above does have harmonic measure of log_{10} 2. There are other ways of enlarging the collection of measurable sets, for example zeta density lim{s --> 1 (from above)) (s-1) sum_{n in A} 1/n^s One can show that any set that has harmonic density or natural density also has zeta density (and is the same) Victor On Fri, Dec 11, 2009 at 3:05 PM, Cordwell, William R <wrcordw@sandia.gov>wrote:
I'm told that, assuming that it is possible to pick a positive integer at random, that one can derive some interesting (and false) consequences, such as 3/8 = 1/2.
Does anyone know of such?
Thanks, Bill C.