From: utzoo!decvax!harpo!seismo!rocheste!FtG
Newsgroups: net.math
Title: 1=/= .999...
Article-I.D.: rocheste.522
Posted: Thu Jan 27 09:46:35 1983
Received: Sun Jan 30 11:27:16 1983
References: hou5e.216

People are missing the whole idea in "proving" that 1=.999... .
We are dealing with infinite sums here, which have very peculiar
properties.
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 proof is too subtle and complex to give over the
net.  Clearly the manipulation of infinite sums is VERY DIFFERENT from
finite ones. Infinite series cannot be capriciously added, multiplied, etc.

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. The mistake people make is in subconciously thinking that the
series and its limit are the same for all practical purposes and
therefore equivalent. REMEMBER: one is a series and the other is a number.
Don't mix apples and oranges.
				Enuf said-
				FtG (rochester)