From: utzoo!decvax!ucbvax!wildbill Newsgroups: net.math Title: Re: = .9999 . . . ? - (nf) Article-I.D.: ucbvax.698 Posted: Tue Jan 25 16:16:34 1983 Received: Fri Jan 28 04:20:40 1983 How about: .999... = 9e-1 + 9e-2 + ..., an infinite geometric series. The formula for the sum of an infinite geometric series is s = a / (1 - r), where a is the first term, r is the ratio between any two adjacent terms, and abs(r) < 1. Applied to this case, with s = 9e-1 and r = 1e-1: s = 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1. Now, if he wants you to prove s = a / (1 - r), use the same trick as in your second proof on the series a + ar + ar^2 + ar^3 + ... (multiply by r and show that most of the terms match up). If he doesn't buy that, you will have to resort to the definition of a limit, which, after all, is what this is. Just convince him that given any number which he wishes to name, you can find a number N such that all numbers of the form .99999...9 with n or more digits differ from 1 by less than the number he gave you, then point to the definition. If he refuses to believe this, he is either hopeless or is not talking about real numbers, whose formal definition is in terms of equivalence classes of sequences of rational numbers. I will be happy to supply more details to interested parties. To find out which he is, ask him whether Achilles can catch the tortoise. Bill Laubenheimer ucbvax!ernie:wildbill ucbernie.wildbill@berkeley