From: utzoo!decvax!ucbvax!CAD:tektronix!zehntel!zinfandel!berry
Newsgroups: net.math
Title: Re: re: .999... =\= 1. - (nf)
Article-I.D.: zehntel.735
Posted: Fri Feb  4 01:18:00 1983
Received: Sun Feb  6 07:55:44 1983

#R:rocheste:-57800:zinfandel:7700002:000:1052
zinfandel!berry    Feb  2 11:08:00 1983

I have not yet been able to read up on 'Cesaro summability', but I hotly
(FLAME ON?) contend that the sequence

	1/2 - 1/2 + 1/2 - 1/2 ... + 1/2*(-1)**(n-1) + ...

has a limit IN THE GENERALLY ACCEPTED SENSE.  let s-sub-n denote the sum 
of the first n terms, and then s-sub-2n is 0, and s-sub-(2n-1) is 1/2, for
any n.  then

	  lim   s
	n -> oo  n

does not exist, when 'lim' is as defined in chapter 1 of all 1st year
calculus books (well maybe chapter 2), you know, the epsilon-delta stuff
someone has summarized well.

	My reference is "Higher Mathematics for Engineers and Physicists"
by Sokolnikoff and Sokolnikoff, McGraw Hill, 1941, page 5.  They examine the
series 1 - 1 + 1 - 1 ...  which is just like the one under discussion,
multiplied by 2.  

	I'm gonna look up that Cesaro reference, you betcha, and I will 
report what I find.  If I'm wrong, I'll admit it, but I bet given the
Math 1 definition of limits, that sequence doesn't converge, nohow!

  	Berry Kercheval
	Zehntel Inc.
	(decvax!sytek!zehntel!zinfandel!berry)
	(415)932-6900