This problem is to connect with algebra of limits . This Can be answer using algebra of limits . In the theory of limite , Lim (1/n) --> 0 . That didn't mean that Lim (1/n) = 0 . However in the actual limite there is small delata value say D such that Rang [0-D, 0+D] \ {0} = Limite poinete .

Thank you Mr. Manoj Sithara for your ideas,

Please check my second point in this thread.

I agree with you that those “infinite related number forms of 1/n--->0, 1/(n+1) --->0, 1/(n+2) --->0, 1/(n+3) --->0,… ” in Infinite Harmonic Series just “get closer to 0 but never reach 0”.

Sincerely yours,

Geng OUYANG

Please, read any introductory book about real analysis. I would suggest the first two chapters of "Understanding real analysis" 2ed from Abbott:

https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116/ref=pd_sbs_14_1?_encoding=UTF8&pd_rd_i=1493927116&pd_rd_r=140VSGAQ75H36WR68KWN&pd_rd_w=QrjoR&pd_rd_wg=y8nW7&psc=1&refRID=140VSGAQ75H36WR68KWN&dpID=41Sm90SnhRL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=detail

This will help you formulate more precise questions... that is if you have any questions remaining about the current subject following this reading.

Thank you very much dear Mr. Bruno Martin for your kind offer.

I care for the wildly accepted process of “bracket computing operation” on the items in the “Un--->0 Infinite Harmonic Series” in Oresme's proof in about 1360 by most of modern mathematicians in the world, because it has disclosed the fatal defects in present infinite related science and mathematics theories system.

But the miserable thing is most of modern mathematicians in the world now don’t really understand it and even don’t know at all how many items each greater than 1/2 people can create from the “Un--->0 Infinite Harmonic Series” with the “bracket computing operation”--------- one or many or infinite (getting closer to infinite).

Some can pretend not to see Oresme's proof and run away from it but can most of modern mathematicians in the world pretend not to see it and run away from it?



Yes, harmonic series diverges again and again.

http://mathworld.wolfram.com/HarmonicSeries.html

" But the miserable thing is most of modern mathematicians in the world now don’t really understand it and even don’t know at all how many items each greater than 1/2 people can create from the “Un--->0 Infinite Harmonic Series” with the “bracket computing operation”--------- one or many or infinite (getting closer to infinite). "

Being a divergent series means that given any number N, there is a partial sum that will be greater than this number. Being true for any natural number, that means you can select as many partial sums as you want greater than 1/2 (otherwise, you also would have bounded the Harmonic series which would be a contradiction to the fact that it is divergent and made of strictly positive numbers).

Finally, please, stop embarrassing yourself by speaking of "most of modern mathematicians don't understand this or that"... you seams to be rather alone at not understanding this. Which brings again my suggestion of thoroughly reading any introductory book about real analysis with the goal of developing a precise vocabulary and relearn the concept of limit, convergence and divergence precisely, not just intuitively.

While it is true that generations of mathematicians worked on this concept to bring it to Weierstrass level of delta-epsilon rigour (and even more general notion of convergence in function spaces, topological spaces, etc.), it is more than 100 years pass the time when "mathematicians did not know better"!

Thank you dear Mr. Mohammad Firoz Khan and Mr. Bruno Martin

I am really seriously and very sure what I am doing.

1, this “world challenging conundrum” focuses on the foundation of present infinite theory system as well as the foundations of infinite related number system and limit theory---------whether the Un--->0 Infinite Harmonic Series is convergent or divergent or something else is another issue and will be discussed in the future.

2, most of modern mathematicians in the world have accepted Oresme's “bracket computing operation” proof for the divergence of Un--->0 Infinite Harmonic Series very well --------the very “bracket computing operation” is “correct and scientific” in present infinite theory system. Isn’t it true?

3, as one of the members in suspended infinite related paradox family, the very “world challenging conundrum” is:

How many items each greater than 1 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... can be created from the Un--->0 Infinite Harmonic Series by the very “bracket computing operation” in Oresme's “correct and scientific” proof-------only one or many or infinite (as many as you like or getting closer to infinite)?

Thank you very much for your understandable ideas and suggestions dear Mr. Marcel Van de Vel.

We debate on some ideas in RG and this made us understand each other more and I take you as one of my honesty friends just because you talk honestly each time.

I think the differences between us are: I think I discovered an “infinite related paradox symptom complex over the sky of present mathematics and science” but you believe “absolutely not”, I turn to philosophy and use many philosophical languages to challenge “classical present infinite related philosophy and mathematics theory systems” and you state firm and enjoy the “non-paradoxical peaceful mathematics”, I have to fight with “fundamental defects disclosed by paradox families” and you have to guard the “perfect present infinite related philosophy and mathematics theory systems”.

On the other hand, my English is really very poor and I have to depend on dictionary very often and sometime very difficult to decide which word to choose when I am doing all the translation work. Pardon me please.

Thank you again my friend dear Mr. Marcel Van de Vel.

Sincerely yours,

Geng OUYANG

I have been away from this discussion thread for a long time, but here is a comment that I have:

Geng, you wrote "1, most of modern mathematicians in the world are very sure to create infinite items each greater than 1/2 and change the Un--->0 Infinite Harmonic Series into a “Vn ＞1/2” infinite series (with infinite items each greater than positive 1/2) in Oresme's proof in about 1360, but is it still that sure that people are able to create infinite items each greater than 1 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from the Un--->0 Infinite Harmonic Series and change the Un--->0 Infinite Harmonic Series into a “Vn ＞1000000000000000000000000000000” infinite series (with infinite items each bigger than any positive constant such as 1000000000000000000000000000000):

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1 + 1+ 1 + 1 + 1 + 1 + 1 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 100 + 100 + 100 + 100 + 100 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1000000000000000 + 1000000000000000 +1000000000000000 + 1000000000000000 + …--->∞"

This is a horrid run-on sentence, and I do not understand what you are asking. ResearchGate tells me that you originally asked this of me, but I have no way to answer such a question, because I thin you did not ask what you intend to ask, mainly because I cannot understand what you wrote. I can comment on some snippets and fragments, however... I am willing to address them, but only one at a time, until I have a clear formulation of your question before me so that I can contemplate your meaning and give you a reasonably well-considered response.

You use the phrase "infinite items each greater than 1/2", but this is ambiguous, if not tautological and devoid of information.

(a) the ambiguity: "infinite items," versus "infinitely many items"...

(b) Once made unambiguous and precise, your use of the phrase "infinite items each greater than 1/2" is either tautological and devoid of information, or leads to a useless debate.

If x is an "infinite item", then by definition of the word "infinite", x is greater than 1/2. If instead of "infinite items", you meant "infinitely many items", then either you are objecting to the axiom of infinity in ZF set theory and its use among modern mathematicians, or you do not understand that there already are infinitely many items, each greater than 1/2, namely those items which are integers that are greater than or equal to 1. In the case in which you are objecting to the use of the axiom of infinity in ZF, I do not plan to debate with you the truth of axioms in a formal system, because this has been debated well and long, and we would only be wasting our efforts. In the case that you do not understand that there already are infinitely many items, each greater than 1/2, AND you accept the axiom of infinity, then you are using an inconsistent reasoning system, and I cannot debate someone who refuses to use a reasoning system that is not known to be inconsistent.

Dear Mr. Matt Insall， thank you！

I am very sorry that you are unable to understand my idea in the question because of my poor English.

I hope you can understand and answer following simplified question:

Are the followings “logical and correct” in present mathematics?

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1 + 1+ 1 + 1 + 1 + 1 + 1 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 100 + 100 + 100 + 100 + 100 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1000000000000000 + 1000000000000000 +1000000000000000 + 1000000000000000 + …--->∞

…

---------Yes or No and why?

Sincerely yours,

Geng OUYANG

For your first inequality, the right way to write it would be to put on the right side something >= term by term to the left side, which would give:

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1/2 + 1/2 + 1/3 + 1/4 +...+ 1/n + …

The way you wrote it, the "partial sums" on each side of your inequality would be false after a few terms! So if your "..." mean that it should be true for any "n" value, that's simply false. If you mean anything else, please by all means, explain it in a more precise notation, à la "delta-epsilon" if you need it, so everybody understands it clearly.

Dear Mr. Bruno Martin，thank you！

Please check the well acknowledged Oresme's “strict mathematical proof” of 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + …--->∞.

Sincerely yours,

Geng OUYANG

Please, look at page 2 of the following document writing Oresme proof correctly by using partial sums:

http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf

Finally, we ALL know the harmonic series diverges, what is are you trying to do anyway?

Thank you for your idea dear Mr. Bruno Martin，

1， The idea of Oresme's proof is “actual infinite--potential infinite” related modern limit theory.

2， The thousands years suspended black cloud of “infinite related paradox families” over the sky of present mathematics and science are produced by the fundamental defects of “actual infinite--potential infinite” in present mathematics.

3， My work care for “the idea of Oresme's proof-------the fundamental defects of modern limit theory”.

4, Whether the Harmonic Series is convergent or divergent or something else is another issue and will be discussed in the future.

Sincerely yours,

Geng OUYANG

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

2，On the Quantitative Cognitions to “Infinite Things” (II)

3，On the Quantitative Cognitions to “Infinite Things” (III)

4 On the Quantitative Cognitions to “Infinite Things” (IV)

5 On the Quantitative Cognitions to “Infinite Things” (V)