Re: infinitely many nn's = infinite nn's?

On Wed, 14 Mar 2007 02:18:17 -0500, herbzet <herbzet@xxxxxxxxx> wrote:


However, do you think, YOURSELF, that infinitely many FINITE values
REALLY CAN sum to a finite value?

I have no opinion on the matter. What I, myself, think is that

a) The n'th member of the sequence 1/2, 1/4, 1/8, ... is
1/(2^n), a rational (finite) number for every (positive)
integer n.

b) The sum of the first n members of the series 1/2 + 1/4 + 1/8 + ...
is 1 - 1/(2^n), a rational (finite) number for every (positive)
integer n.

c) The limit of 1 - 1/(2^n) as n increases without bound is 1:
that is, 1 - 1/(2^n) gets as close to 1 as you like, for
all n that are large enough, and

d) 1 is unique in this respect.


Right. Though the following seems to be even clearer:

Consider the infinite many "finite values": 0.1, 0.01, 0.001, ...

Now the sum of these values

0.111...

is obviously a finite "value". ;-)

You know, the sequence 1/2, 1/4, 1/8, ... is already close to the edge: the
sequence 1/2, 1/3, 1/4, ... does not "sum up" to a "finite value" any more.


Whether or not the sum of an infinite number of of finite values
can sum to 1 I don't know.

Though we actually SPEAK of an "infinite sum" in this context (defined in
the way you described, of course).


I don't know if "the sum of an infinite number of finite values" even
makes sense. It doesn't make sense to me (but I'm not an expert).
Perhaps it can be given a sense, but I don't need that.

See, comment above.


I can work perfectly well with 1 being the _limit_ [...]
of an infinite series [...]

Right.


Let's put it this way: Suppose you have a piece of rope that
is one meter long, and you cut it in half and put one of the
halves in a box. Then you cut the piece in your hand in half
and put one of those halves in the box. Then a third time, and
a fourth, and so on. What is the sum of the lengths of rope
in the box as you make more and more cuts? Their sum approaches
one meter, the more cuts you make, obviously. What is their
sum if you make an actual [countable] infinite number of cuts?
I don't really know, or need to know. If you have a good answer,
so much the better.

Intuition suggests that we will not exceed the end of the rope by cutting
"halfway points" even if we consider [countable] infinite many such cuts.
On the other hand it's clear that we cannot state any length < 1 m such
that our cuts do not exceed that length. So... Hmmm...


F.


P.S.
It's a pity that I know almost nothing about Nonstandard-Analysis. But I
could imagine the following situation: the countable infinite pieces only
sum up to 1 - an infinitesimal length (since the cuts _never_ actually
reach the end of the line, i.e. the 1 m point, while on the other hand
there's no finite, i.e. not-infinitesimal, distance d such that our cuts do
not exceed 1 - d).

--

E-mail: info<at>simple-line<dot>de
.

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