From: Robin Chapman Subject: Re: An infinite product Date: Wed, 12 Jul 2000 18:12:48 GMT Newsgroups: sci.math Summary: [missing] In article <396C82D7.409F4EC7@stat.fsu.edu>, George Marsaglia wrote: > The probability that a random n x n binary matrix will be nonsingular > (all arithmetic mod 2) involves the product > > (1-1/2)(1-1/4)(1-1/8)...(1-1/2^n), > > which quickly approaches .288788095086602421278899721929... > > Can the limit be expressed in terms of familiar numbers or elementary > function values? It can be expressed in a "closed" form involving the Dedekind eta function, but at that is (essentially) the product (1 - x)(1 - x^2)(1 - x^3) ... that really isn't that informative. Better value is the Euler pentagonal number theorem which states that (1 - x)(1 - x^2)(1 - x^3) ... is the sum of (-1)^k x^(k(3k+1)/2) over *all* integers k. Using this greatly speeds up numerical computations like the above case of x = 1/2. -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy.