Re: * says: Definition: sum{i in N} i = 0

On 8 Jun., 14:58, Franziska Neugebauer <Franziska-
Neugeba...@xxxxxxxxxxxxxxxxxxx> wrote:
WM wrote:
On 8 Jun., 10:34, Franziska Neugebauer <Franziska-
Neugeba...@xxxxxxxxxxxxxxxxxxx> wrote:

I think you should know that a divergent positive series cannot
have the sum 0.

A divergent series does not possess a defined value within the
framework used to determine its divergency at all.

A divergent series has the property of being divergent, which is but
another way of saying that it has no finite value

This is not correct. If have learned something like that:

http://en.wikipedia.org/wiki/Divergent_sequence

Therefore "divergent" simply means there is no L e M which is _limit_
according to the usual rules of limits.

Then you should improve your knowledge: Even it there is no L in M, we
can formulate the limit of a divergent series. This is a
generalization of the usual sum, but a consistent generalization (in
contrast to ***'s definition). A sequence (a_n | n in N) of positive
terms a_n has the (improper) limit oo, if the sequence (1/a_n) exists
and has the limit 0. A series can be treated as the sequence of its
partial sums.

The value or sum or limit of a series is the limit of its sequence of
partial sums and that is an extended definition of the sum of its
terms.

You or *** may (for whatever purpose) hence define ad lib which is
perfectly legal. Whether it is useful is a another issue.

which in turn implies that is has not the value 0

non sequitur.

"Divergent series" means: There is no limit, sum, value in the real
numbers. Hence ***'s definition is a contradiction with the fact that
1) every natural number is positive and
2) that a sum of naturals is larger than any of its summands and
3) that already the sequence of naturals is divergent.

which implies that ***'s definition is wrong

non sequitur. (Euler/Ramanujan et al. would be "wrong" either).

They are wrong. Would you also agree to Eulers result

1/(1-2) = 1 + 2 + 4 + 8 + ... = (-1) > 1?

after the divergence has been shown. One way to do
this is to start with Nicole d'Oresme.

Another is to stop with him right now.

You do not know of the technique to use a majorante of a sequence?


It has already been in depth explained to and ignored by you that
the diagonal number has a convergent series, i.e. that its limit exists
and that the limit is not in the list.

Where does its limit exist? If it does not exist in the list, then it
does not exist at all, because: why does it not exist in the list, if
it could exist? Compare the tree: Removing the limits (infinite paths)
does not change anything in the tree.

Here is an example which might enlighten you: Consider the list of
decimal approximations of pi:

Of what?

Of the fact that Cantor's diagonal argument fails for real numbers.

Regards, WM

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