Re: infinity

In article <MPG.1d63f7bee9f2480698a029@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1d62d4357285553c98a015@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >
> > > David R Tribble said:
> > > > Tony Orlow (aeo6) wrote:
> > > > >> Does the infinite series (10,-1,10,-1,10....) converge to zero?
> > > > >> Do the terms have a limit of zero? No, sorry.
> > > >
> > > > David R Tribble said:
> > > > >> Well, I can rearrange the terms so that they add to zero:
> > > > >> s = 10 - 1 + 10 - 1 + 10 - 1 + 10 - 1 + 10 - 1 + ...
> > > > >> s = (10 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1)
> > > > >> + (10 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1)
> > > > >> + ...
> > > > >> s = (0) + (0) + (0) + ...
> > > > >> s = 0
> > > >
> > > > Tony Orlow (aeo6) wrote:
> > > > > Gee, you seem to be left over with a whole lot of extra +10's, don't
> > > > > you?
> > > >
> > > > How so? There is an unending supply of 10s, as well as an unending
> > > > supply of 1s. I don't seem to be using any terms that don't already
> > > > appear in the original series. If I do end up with "extra" terms,
> > > > how many extras are there?
> >
> > > Nine extra unused 10's for each of your zero-sum terms. Cantorians seem
> > > to
> > > think they can push anything extra away to infinity, and then they don't
> > > exist.
> >
> > Not at all!
> >
> > Each extra 10 needs only to be pushed out finitely far, until 10 unused
> > -1's can be accumulated. According to TO's own claimed rules on
> > commutativity, this is quite proper. Since there are infinitely many
> > 10's and infinitely many -1's, there is no problem.



> No problem because you push them all away.

But it was TO himself who stated that one could, using commutativity,
rearrange these things anyway one liked.

SO either TO must retract that statement or live with the consequences
of it.

It is only in three cases that order is totally irrelevant:
(1) absolutely convergent series;
(2) series in which either the number of negative terms
or the number of positive terms is finite.
(3) series in which the subseries terms of one sign converge
and the subseries of terms of the opposite sign diverge

In the remaining case, in which both the subsequence of all positive
terms and the subsequence of all negative terms diverge, rearranging
terms can alter the results.
.

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