From: utzoo!decvax!duke!harpo!seismo!hao!menlo70!sytek!zehntel!tektronix!tekmdp!laurir
Newsgroups: net.math
Title: Re: 1=/= .999...
Article-I.D.: tekmdp.1735
Posted: Fri Jan 28 22:58:38 1983
Received: Thu Feb  3 02:07:37 1983
References: rocheste.522

	Example consider S = 1/2 - 1/2 + 1/2 -1/2 .... .
	Using the arguments given by some people, I can prove that
	S is both 1/2 and -1/2! The "true" answer (as any calculus text will
	tell you) is that S = 1/4!

The "true" answer (as any calculus text will tell you) is that the
sequence S does not converge, and so is not "equal" to any particular
number.

	Getting back to what is .999..., the first mistake everyone is
	making is: 1/9 = .111.... ! This is not true, strictly speaking.
	The infite series .1 + .01 + .001 + ... CONVERGES to 1/9.
	It is not really 1/9.  Hence 9 times a number that CONVERGES
	to 1/9 is a number that CONVERGES 1.

This is just a matter of agreeing on notation.  The majority of those
who are so interested in mathematical trivia that they didn't "n"
this article seem to agree that ".999..." is just shorthand for

	 inf
	sigma  (9*10^-i)
	 i=1

which is the limit of the sequence, which *is* a number, equal to 1.
Hence, 1/9 = .111... and 1 = .999...

  -- Andrew Klossner (decvax!tektronix!tekmdp!laurir)