From: bruck@math.usc.edu (Ronald Bruck) Newsgroups: sci.math Subject: Re: infinite sums and products Date: Thu, 29 Oct 1998 10:02:24 -0800 In article <71a7ou$l1v$3@clarknet.clark.net>, Harlan Messinger wrote: :Markus Deserno wrote: :: Harlan messinger wrote ::> Define a_n = 0.1 for n = 1, 2, 3, .... ::> ::> s[a] approaches infinity ::> p[a] approaches 0 : :: No, not true. You did not read my posting correctly: : :: s[a_n] = 0.1 + 0.1 + 0.1 + ... --> infinity :: p[a_n] = (1 + 0.1) * (1 + 0.1) * (1 + 0.1) * ... --> infinity : :Sorry, you're correct. Doesn't matter, though. : :Define a_n = -0.1 for n = 1, 2, 3, .... : : s[a] diverges to -oo : p[a] converges to 0 : :Define a_n = -1 for n = 1, 2, 3, .... : : s[a] diverges to -oo : p[a] IS 0. It's for this very reason that convergence of infinite products is defined differently. That is, one says that an infinite product of p(n) converges if, and only if, the sequence of partial products converges to a NON-ZERO number. This is to make the convergence of \product_n p[n] is equivalent to the convergence of \sum_n log p[n]. When p[n] = 1 + a[n], one has inequalities a[n]/(a[n]+1) \le log(1 + a[n]) \le a[n] (for a[n] > 0). and from this it's easy to see that \sum_n log(1+a[n]) converges iff sum a[n] does. I repeat: the definition is NOT that the partial products converge. --Ron Bruck -- --Now 800% ISDN from this address (2B channels + STAC compression)