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

On 10 Jun., 15:52, Franziska Neugebauer <Franziska-
Neugeba...@xxxxxxxxxxxxxxxxxxx> wrote:
WM backlog:

A

"Divergent series" means: There is no limit, sum, value in the real
numbers.

"No limit" precisely means there is no L in M having the properties
given inhttp://en.wikipedia.org/wiki/Divergent_sequence.
"Divergent series" means: the series has no limit.

Accept or deny?

Of course accepted, although you probably should not gather all your
mathematical knowledge from Wikipedia alone, in particular not from
the German Wikipedia. If you add the fact that the limit of a series
is the sum of its terms, then you get the result that a divergent
series has no sum in R. It is easy to prove that it cannot have a sum
in R. Therefore it is easy to prove that ***'s definition is wrong.

B

***'s definition (assignment of the value 0 to a non-convergent
series)does neither contradict to

1) every natural number is positive, nor
2) that a finite sum of naturals is larger than any of its
summands, nor
2a) that an infinite sum of naturals (> 0) has no limit in N, nor
3) that the infinite sequence of naturals is divergent in N.

This shows: You have no clue of logic conclusions.

If you want to claim that ***'s definition does contradict to 1),
2), 2a) or 3) you have show that.

If you believe to be the chambermaid of Hilbert's hotel, then I cannot
convince you, by any proof, that this hotel does not exist. Therefore
I abstain from doing so.





WM wrote:
WM wrote:
On 9 Jun., 10:44, Franziska Neugebauer <Franziska-
Neugeba...@xxxxxxxxxxxxxxxxxxx> wrote:
WM wrote:

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.

This is certainly a possible assignment but it is no longer in the
original range of the series (same applies to H&J).

It is completely in the original range.

No. "oo" is certainly _not_ in the original range c= N.
There is no element named "oo" in N or R.

But 0 is certainly in this range, and the method of reciprocals is
allowed.

Read what you wrote! You had written

| 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.

I said that

"oo" is certainly _not_ in the original range c= N
because there is no element named "oo" in N or R.

Now you wrote

| But 0 is certainly in this range, and the method of reciprocals is
| allowed.

1. This is not an argument at all since it is not 0 but you claim "oo"
to be the "(improper) limit" of a divergent series.

And I claim that this limit validly can be calculated by the
reciprocals. The reciprocals have the limit 0 which is in R.

2. In R there is no reciprocal of 0. It is meaningless to say "oo" were
the reciprocal of 0 in the context of R and you know that only too well.

Concerning limits it is correct to say that a sequence has the
(iproper) limit oo if the sequence of reciprocals has the limit 0. The
only condition required here is that no term of the sequence is 0,
because then the reciprocal does not exist in R. But this condition is
satisfied for the sequence of partial sums of N.

At least, I teach this so. Am I in error?





A series can be treated as the sequence of its partial sums.

The value of an infinite series _is_ _defined_ as the limit value
of its partial sum. If there is no L in M there is no such limit.
In this case we say: The infinite series does not converge.
Period.

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.

? (possibly too many or's)

Rather too less,

Keep it Simple, Sweatheart (KISS).

Oh!

Here is, as I think, the generally accepted definition of convergence
(Grauert, Lieb: D+I vol. 1,p. 41)

Konvergiert eine Folge (a_nu) gegen x_o, so nennt man x_o Grenzwert
(Limes) der Folge ...
Eine nicht konvergente Folge heißt divergent.

(Grauert, Lieb: D+I vol. 1,p. 48): Mit Hilfe des Grenzwertbegriffes
ist es möglich, in gewissen Fällen auch unendlich vielen reellen
Zahlen eine wohlbestimmte Zahl als Summe zuzuordnen.

Obviously, in other cases, this is impossible, isn't it?

To what particular question do you reply to by this particular quote?
This particular quote you intent to support _which_ particular
statement?

I reply to KISS: "The value or sum or limit of a series" and to the
denial of the interpretation of the limit of the series as being the
sum of its terms, as one reads it in modern text books (of low
quality).

Would you also agree to Eulers result

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

Euler's/Ramanujan's result actually is:

1 + 2 + 3 + ... = -1/12 (R)

So I would like to see your calculation _step_ _by_ _step_ which
leads to your statement (*). Then I can demonstrate your error.

How would you accomplish that?

By identifying your error.

By your usual unlogical and wilful commenting?

No. By identifying your error as usual.

An error cannot be demonstrated *by you* in my statement, because
A) 1 + 2 + 4 + 8 + ... = (-1)
is the same nonsense as
1 + 2 + 3 + ... = -1/12.

I am going to identify the error in your _derivation_ of (*) after
you will have presented it.

B) The statement is not mine but is a result of Euler.

The statement of Euler/Ramanujan is

1 + 2 + 3 + ... = -1/12 (R)

Your statement under discussion (*) is

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

What now?

I would like to see _your_ calculation _step_ _by_ _step_ which
leads to your statement (*). Please do or otherwise scrub (*).

If -- what I do not know -- Euler has _literally_ written (*) you may
name a reference or even better quote his derivation.

I think you can find this in many books on history of math. I
recommend M. Cantor. (What I have stored in my memory isn't always
labelled by a reference, nevertheless in most cases it is correct.)

I would not rely on that.

But you would rely on the axiom of choice! LOL.

Regards, WM

PS: I have found a reference: W. Mückenheim: Kleine Geschichte der
Mathematik - von den Anfängen bis ins 18. Jahrhundert (Skriptum zur
Vorlesung), Augsburg 2001, p. 110.

There are two issues which have aggregated now:

1. By writing

| Would you also agree to Eulers result
| 1/(1-2) = 1 + 2 + 4 + 8 + ... = (-1) > 1? [(*)]

_you_ impute that Euler literally has written (*). An accepted proof
that a person A has literally written B is _quoting_ literally from the
work of A.

If I wrote a scientific article about this result of Euler's (which
was also derived by Wallis, if I am right) then I would have studied
his work. But I was only en passant mentioning an absurdity fitting
well to another absurdity.

You did not provide us with such proof.

If Euler's result is wrong, you should be able to point that out. If
you accept Euler-Ramanujan, however, I do not see why you disagree
with Euler alone.

1 + 2 + 3 + ... = -1/12 and
1 + 2 + 4 + 8 + ... = (-1) fit very well together.

On the other hand we could conclude, if
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ... = -1/12, then
1 + 2 + 4 + 8 + ... =< -1/12 - 3 - 5 - 6 - 7
- ... =/= (-1)
unless
-1/12 - 3 - 5 - 6 - 7 - ... = (-1).
But why not? In principle, if *** is right (as you seem to believe)
then everything could be (-1). That would considerably simplify
mathematics. We would no longer need computers! Students would always
pass the exams. There would be no drop outs!

There remains only one problem, the last unsolved problem of
mathematics: This mathematics cannot be applied for calculating Grocer
bills and bank accounts. But that does not matter, "modern
mathematics" finally has completely occupied the victory rostrum.

I do not judge whether your referenced own work literally quotes
the work of Euler. I will surely not read it in order do unburden you
from giving the appropriate quote _here_. You who claim that Euler has
literally written (*) have to prove it _here_ (in sci.math).

2. To my best knowlegde Euler/Ramanujan are assumed to have written

| 1 + 2 + 3 + ... = -1/12 (R) [(E)]

and not (*). So to get to (*) <-> (E) you need some kind of formula
manipulation which you have not presented at all. My claim is: Present
the manipulation that yields (*) <-> (E) and I will show your error
or retract my claim if I can't find an error.

As far as I know, Euler simply applied the well known formula 1/(1-q)
without paying attention to convergence, as was usual at those times
(although Euler was the first to mention a convergence criterion.
(You can find it on p. 10 of my "Die Mathematik des Unendlichen". Here
it is off topic.) Leibniz and Jakob Bernoulli (1696) also agreed to
1/2 = 1/(1-(-1)) = 1 - 1 + 1 - 1 +-... = 0 + 0 + 0 + ... = 0. With
***'s attitude of mathematics, we return to those times where the monk
Grandi explained the creation of the world from nothing by just this
effect: If you add enough zeros, then the result is 1/2.

Would not ***'s and your modern mathematics support just this position
for actually infinitely many zeros?

0 + 0 + 0 + 0 + ... = 1/2

Of course. This must have been known to God already. There is no other
way to create something from nothing. Or...???

If we take an empty set, { }, {{ }}, ... Perhaps God created the world
of empty sets?????

I do not judge whether your referenced own work contains such
manipulation. I will surely not read it in order do unburden you from
giving the appropriate quote _here_ (in sci.math).

To sum up: Until you perform your duties your claim Euler has written
(*) is vacuous.

I will bear this verdict.

Regards, WM

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