Re: Tea cups and elephants

On Nov 30, 7:29 am, matt271829-n...@xxxxxxxxxxx wrote:
On Nov 30, 2:14 pm, Randy Poe <poespam-t...@xxxxxxxxx> wrote:



I think the problem may also be in insisting that
the sum "in the limit" can be written as something
involving plus signs, even when there are an
infinite number of terms "in the limit".

By "0 + 0 + 0 + ..." I mean the limit as n->oo of the sum of n zeroes.
I am assuming that this is a well-defined notion, and that it equals
zero.

Don't assume it. Use the definitions.

The infinite sum sum_i=1^oo a_i
also written a_1 + a_2 + a_3 + ...

is defined to be the limit (if it exists) of the sequence

a_1, a_1 + a_2, a_1 + a_2 + a_3, ...

called the sequence of partial sums. So
0 + 0 + 0 + ...
means the limit (if it exists) of the sequence
0, 0 + 0, 0 + 0 + 0, ... which is the limit of
0, 0, 0, ... which is 0. That is the only meaning for
0 + 0 + 0 + .... You are not actually adding up infinitely many zeros
(which I believe you understand).
0 + 0 + 0 + ... does not "approach" 0. It is 0.
1/2 + 1/4 + 1/8 + ... does not "approach" 1. It is equal to 1. (I also
believe that you understand that).




What is
actually happening is that there is, as I wrote,
a sequence of numbers. When you consider the numbers,
the value of each finite term, I hope your intuition
doesn't still suggest that {0,0,0,...} should
"eventually" reach 1.

It seems clear to me that 0 + 0 + 0 + ... (as defined above) must
equal zero. On the other hand, it also seems clear that sum_i=1^n c/n
approaches 0 + 0 + 0 + ... as a limiting case, so that when we find
the limit as n->oo we actually *do* get 0 + 0 + 0 + ...

No. That's an optical illusion. What do you mean to say that something
(in this case the sequence {sum_i=1^n c/n} ) "approaches" an infinite
series? "Approaches" presupposes the notion of limit and limit
presupposes some way of measuring distance or closeness. How would you
measure the distance between the members of the sequence {sum_i=1^n c/
n} and the infinite series 0 + 0 + 0 + ... ? It seems to me that
there are two possible ways:

1) Since the definition of an infinite sum is the limit of the
sequence of partial sums (see above) we could insist that the two
sequences approach each other component-wise -- the first member of
the sequence approaches the first partial sum, the second member of
the sequence approaches the second partial sum, .... Let's see: Your
sequence evaluates to
{1, 1, 1, ...} and the sequence of partial sums evaluates to
{0, 0, 0, ...} (see above). Doesn't look like there is any chance for
an approaching using this method.

2) What is an infinite sum? It's the limit (if it exists) of the
sequence of partial sums. What is a limit? It's a (in these cases)
real number. So an infinite sum (if it exists) is a real number.
Likewise the limit of your sequence {sum_i=1^n c/n} is a real number.
We could say that the sequence approaches the series if those two real
numbers were the same. You have already pointed out that they are
different, so that method won't work.

What you are trying to say is that the sequence
c, c/2 + c/2, c/3 + c/3 + c/3, ... looks like it is approaching
0 + 0 + ... because you have a number of things added together and the
individual summands are approaching 0 and the number of summands is
going to infinity. As I (and others) said, that's an optical illusion.
c/4 + c/4 + c/4 + c/4 is a real number. The number c. c/4 + c/4 + c/4
+ c/4 is just another way of writing or describing the number c. It's
a string of symbols. Distance and hence limit and hence "approaches"
is not defined for strings of symbols but for real numbers. Until
someone defines a way of measuring the closeness of two strings of
symbols it is just without meaning to say that "sum_i=1^n c/n
approaches 0 + 0 + 0 + ... ". You are mixing up your categories.

You are not alone. Most people are fooled by this optical illusion
until they do the math. I believe that there are some famous
"paradoxes" in probability based on this optical illusion.

Perhaps it is because in the past few hundred years the foundations
mathematics, at least as far as the real number system is concerned,
have been rigorously axiomatized that we can discriminate between the
optical illusions and the solid math and use the solid math to create
models of reality which (for better or worse) is one of the bases of
our technological civilization while philosophers who refuse to define
their terms have been arguing over the same things for thousands of
years. And of course all dolphins ever do is muck around in the water
having a good time of it.

I presume that
this is wrong because lim n->oo (sum_i=1^n c/n) is clearly equal to c.
However, I can't quite grasp why we get the wrong answer. Do you see
what I'm getting at?

Note that I am not assuming that the terms actually equal zero for any
finite n, any more than I believe lim n->oo (c/n) = 0 implies c/n = 0
for some n. In neither case is anything evaluated with n "set to
infinity".

Well, you ARE saying that the limit of your sequence is a series with
infinitely many summands...
.

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