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

WM wrote:

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.

I either do not gather it from third-rate literature recently referred
to by you.

If you add the fact that the limit of a series
is the sum of its terms,

There is a subtle difference in the correct wording: The value of an
infinite series is _defined_ to be the limit of the partial sums of its
underlying infinite sequence _if_ the limit exists. Otherwise there is
no value. In that case the (value of the) series is undefined.

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.

You are again in error showing that you do not really accept the
definition in wikipedia: The correct wording is: A divergent series has
no value (= is undefined) if there is no limit of the partial sums of
its underlying sequence. Independently from the limit definition the
divergent series has no value on its own right unless one defines its
value.

Therefore it is easy to prove that ***'s definition is wrong.

From the correct definition it follows that everybody is free to define
the value of the divergent series ad lib.

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.

Let me assure you that I would never ever check in into Hilbert's hotel
not even in room #1. LOL Nonetheless you have to prove your claim.

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.

Irrelevant. BTW: 9 lines below I'll present the calculation of the
reciprocals.

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.

***'s definition involves the divergent _series_

sum_{i e N} i,

i.e.

lim sum_{i=0}^n i
n -> oo

The sequence of reciprocals of the partial sums of his series is

(1 / sum_{i=0)^n i)_{n e N}

Already the first value (n = 0) of this sequence of reciprocals is
undefined. OK nitpicking. Let's drop it for simplicity. According to
you we shall now define, if I get you right:

1
lim sum_{i=0}^n i := ------------------------- = 1 / 0
n -> oo lim (1 / sum_{i=0)^n i)
n->oo

Q: How do we call the value of "1/0"?
A: "1/0" is undefined.
Q: Is "1/0" an element of R or N?
A: No.
Q: Does the method of the reciprocals shed any light on some "intrinsic
value of ***'s sum" which is hidden by the series' divergence?
A: No.
Q: Does it prove ***'s definition wrong?
A: No.
Q: So what?
A: Good question.

The only condition required here is that no term of the sequence is 0,

The first term *has* value 0, unfortunately.

because then the reciprocal does not exist in R. But this condition is
satisfied for the sequence of partial sums of N.

If we want to be ruthless we must concede that you are wrong.

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

Im sorry to tell you that you are in this particular case.

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).

Reihe = series, Folge = sequence. Your quote is not about series at all.

The true story about series and sequences can be read, pardon wikipedia
again, here:

,----[ http://en.wikipedia.org/wiki/Series_%28mathematics%29#Formal_definition ]
| Mathematicians usually study a series as a pair of sequences: the
| sequence of terms of the series: a0, a1, a2, ? and the sequence of
| partial sums S0, S1, S2, ?, where Sn = a0 + a1 + ? + an. The
| _notation_
|
| \sum_{n=0}^\infty a_n
|
| represents then a priori this pair of sequences, which is always well
| defined, but which may or may not converge. In the case of
| convergence, i.e., if the sequence of partial sums SN has a limit, the
| notation is also used to denote the limit of this sequence. To make a
| distinction between these two completely different objects (sequence
| vs. numerical value), one may sometimes omit the limits (atop and
| below the sum's symbol) in the former case, although it is usually
| clear from the context which one is meant.
`----

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.

I meant that I would not rely on your memory.

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.

For the record: WM does not give quote for his claim so his claim that
Euler has literally written (*) is not proved true (yet).

You did not provide us with such proof.

If Euler's result is wrong,

To begin with I doubt that (*) is of Euler's origin, i.e. that he
literally has written (*).

you should be able to point that out.

"(-1) > 1" is mathematically wrong. That does not prove that (*) is by
Euler. I suppose it's a Mückenheim.

If you accept Euler-Ramanujan, however, I do not see why you disagree
with Euler alone.

I assume that (E)

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

is from Euler and/or Ramanujan. And I can't see any mathematically
logically valid path (manipulation) going from (E) to (*).

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

If you write them in two successive lines they are close together but
that does not constiute a derivation from (E) to (*).

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).

(*), which you attribute to Euler, is

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

Please demonstrate only the first step of (A) = (E) on the way to (*).

I calculate as follows

(A, E)
<=> (1 + 2 + 4 + 8 + ...) = -1/12 - (3 + 5 + 6 + 7 + 9 + ...)

What the next step on your way to (*)?

But why not? In principle, if *** is right (as you seem to believe)
then everything could be (-1).

Slow down a bit. It is late and I cannot follow your weird explanations.
You probably cannot redefine at will. But *** simply defined what was
previously undefined.

That would considerably simplify mathematics. We would no longer need
computers! Students would always pass the exams. There would be no
drop outs!

According to you we even don't need resource allocation conflicts when
we accept WM's law of "least injustice". I will suggest you for Nobel
Peace Prize.

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.

The grocers will ask you for calculating their bills. Perhaps the only
kind of maths you'll master.

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,

That sounds less like Leonhard but like Wolfgang Euler.

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.

Irrelevant even if the Pope agreed to it.

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.

irrelevant. evading the issue.

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

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

irrelevant. evading the issue.

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?????

completely irrelevant. evading the issue.

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.

So we will bear that you retract "(*) is by Euler".

F. N.
--
xyz
.