Dear Dr. George Stoica, thank you.

It is well acknowledged that “Limit theory is a very useful cognizing tool in human science.” But, as a typical case, the Harmonic Series Paradox of Zeno’s paradox family members discloses following two defects in the foundation of limit theory:

 1, what kind of “infinite number forms” it treats-------potential infinite number or actual infinite number or missing ups or …?

2, in present mathematical number system, there are no “infinite number forms” at all.

If no one in the world now can self-justificationaly solve the problem (defects) disclosed by “How many items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series in present science theoretical frame?”, we have to admit the fact and meet the challenge and do something sooner or later to make up the defects in the foundation of limit theory.

Yours sincerely,

Geng

In the classical analysis lim(x ---> x0) f(x) = a if and only if forall epsilon > 0 there exists delta > 0 such that for all x with |x-x0| < delta it holds |f(x) - a| < epsilon. In the non-standard analysis the limit is the standard part f(y) where y is a non-standard number with standard part equal x, provided that f is a standard function. Both notions, which are equivalent as proven by Abraham Robinson (and later by Edward Nelson) are very well founded. This is also true for other kinds of limits, like sequence, series, or set sequence limits. It is easy to see that classically the infinities used are potential (for all epsilon > 0 means for arbitrary small epsilon, and that means potentially infinitely small epsilon) and in the modern approach called non-standard analysis infinities are actual (because there exists elements which are infinitely small, and other which are infinitely big, and the infinitely small elements are cut off by the standard part operation). But by the fact that the notions are equivalent and that both are good foundations for the limit theory, it is clear that this distinction between actual and potential infinity is not important for the definition of limit. We do not win any supplementary insight in limits by this particular distinction.

Dear Dr. Mihai Prunescu, thank you.

It is well acknowledged that “Limit theory is a very useful cognizing tool in human science.”

But the trouble is: in front of the Harmonic Series Paradox, both “delta- epsilon related classical limit theory” and “non-standard number related non-standard analysis theory” can do nothing--------they both can not self-justificationaly solve the problem (defects) disclosed by “How many items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series in present science theoretical frame?” just because in present classical infinite related science system (1) the confusing “potential infinite” and “actual infinite” are always mixed up and shifted as one wishes; (2) there are no “potential infinite number” and “actual infinite number” in present mathematical number system.

Yours sincerely,

Geng

Hi!

As a follower received your question. I am not a mathematician. Just surfed, found 2 of your papers and the following information. Hope it is useful.

Basic properties, evaluating limits

http://math.feld.cvut.cz/mt/txtb/5/txe3ba5b.htm

Limit Theorems

http://archives.math.utk.edu/visual.calculus/1/limits.18/

Paul's Online Math Notes The Definition of the Limit

http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

Limits (Formal Definition)

https://www.mathsisfun.com/calculus/limits-formal.html

LIMITS

A sequence of rational numbers

The limit of a variable

Left-hand and right-hand limits

The definition of the limit of a variable

The limit of a function

The definition of the limit of a function

Section 2:

Theorems on limits

Limits of polynomials

http://www.themathpage.com/acalc/limits.htm

Dear Geng, is it so paradoxically? I understand that we speak about Sum (n --> \infty) (1/n). For every position n0 of the bracket "{" there is a position n1 such that for every position n2> n1 of the bracket "}", the part of the series 1/n0 + ..... + 1/n2 > 1/2. I hope that this is what you meant. Now you are asking about the cardinality of the set  of sequences of natural numbers {x1 < x2 < x3 < .....} such that the sums 1/x[i] + ... + 1/(x[i+1]-1) are all bigger than 1/2. In the moment I tend to believe that the cardinality of this set of sequences is the continuum. But even if I am wrong and the cardinality is just the countable infinity, it is pretty clear that the answer is unique and provable. Do we really need the difference between actual in potential infinity in order to formulate and to solve this problem?

Dear Dr. Jetty Ramadevi, thank you for introducing some well acknowledged theories of limit.

Yours sincerely,

Geng

Dear Dr. Mihai Prunescu, thank you.

1, are you so sure that we can really have infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series in present science theoretical frame?

2, according to my studies, all the infinite related paradox cases are challenging us seriously “what are the differences between actual infinite and potential infinite-------- ideas, definitions, number forms, treating ways, …”

Yours sincerely,

Geng

So, if we are speaking about Sum (n --> \infty) (1/n) and about finite sums of the shape 1/x[i] + ... + 1/(x[i+1]-1) for some sequence of increasing natural numbers x1 < x2 < x3 \infty) (1/n)  = \infty. The set of this sequences x1 < x2 < x3

Dear Dr. Mihai Prunescu, thank you.

Are you so sure you are able to prove that “infinite items each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000)?

Yours sincerely,

Geng

If you have some difficulties with infinity in set and limit, refer alternative set theory, which evolved around 90's and talks about theory of natural infinity. http://plato.stanford.edu/entries/settheory-alternative/

Dear Geng, yes, I am sure. Just look:

1/3 + 1/4 > 2 x 1/4 = 1/2

1/5 + 1/6 + 1/7 + 1/8 > 4 x 1/8 = 1/2

1/9 + 1/10 + ... + 1/16 > 8 x 1/16 = 1/2

If you put brackets over bigger intervals at the beginning, all the intervals must increase. But the series is divergent, so ANY rest-series is divergent too, so you will be able to put the first bracket anywhere and you find a finite sub-series starting there, which is bigger than, say, 10^{16}. For the rest of the series you can repeat this. The series being diverging, you have a lot of freedom.

Dear Dr. Panchatcharam Mariappan, thank you for your recommendation..

Yours sincerely,

Geng

Dear Dr. Mihai Prunescu, thank you.

The proof you gave in above post was given by Oresme in about the year of 1360. But it is the modern version of the famous Ancient Zeno’s Achilles--Turtle Race Paradox--------the Un--->0 Harmonic Series can be changed into a Un’--->10000000000 Positive Infinite Series:

1＋1/2 ＋1/3＋1/4＋…＋1/n ＋… =  10000000000＋10000000000＋10000000000＋…

Yours sincerely,

Geng

In my opinion, there is no defect here. The most interesting geometrical meaning of the notion of the limit of a function is the notion of an asymptote. This notion helps in graphical representation of the smooth functions, as well as in the study of analyticity. On the other hand, let me recall the well - known example of a real valued function of one real variable, which is a C-infinity function, without being (real) analytic: f(x) = 0 for x0. Proving that this function is indefinitely derivable on the real axes (moreover, is of C-infinity class), is a question of basic limits at the origin.. All the important results on continuity or differentiability of a function (of one or several real variables) involve limits. If we want to speak about limits involving numbers, the fact that lim(1/n)=0, but lim(1+1/2+...+1/n)=infinity when n goes to infinity, I don't see any contradiction here: there is a clear argument for the latter limit.  

Dear Dr. Octav Olteanu, thank you very much for your frank point of view. The following is my frank point of view:

1, In the suspended Ancient Zeno’s Achilles--Turtle Race Paradox, Zeno proved a fact “no matter how big Achilles’ steps are, he can never catch up with Turtle when the Turtle in the running race” .

Now in our modern mathematics, we “strictly” prove a fact with limit theory “we can produce infinite items each bigger than10000000000000000000 from the Un--->0 Harmonic Series” and change the Un--->0 Harmonic Series into a Un---> 10000000000000000000 Infinite Series.

2, Is Un--->0 = Un--->10000000000000000000 (any constants) in the theory of infinite series in modern mathematics?

Yours sincerely,

Geng

On one hand it is a trivial and well known fact that we will catch the Turtle if we are faster than the Turtle. On the other hand Analysis proves that we catch the Turtle. However, the limit is NOT 1 + 1/2 + 1/3 + .... In the case of Achiles and the Turtle the limit is 1 + 1/10 + 1/100 + ... + 1/10^n + ... in the plausible case that Achiles is 10 times faster than the Turtle. Attention: this is not the harmonic series. This is the geometric series! So again: our experience shows that we catch up the Turtle, and Analysis proves that we catch up the Turtle. So where is Zeno's paradox? 

Zeno's paradox is that it takes infinite time to READ an infinite series, even if the series converges. This is normal because words like "1/10^n" take longer and longer time to be read (with n increasing), while they refer to smaller and smaller time intervals in the infinitesimal process of catching up the Turtle. Happily we introduced shorthand notations for all recursive functions. We can describe the function by the algorithm that produces it, and every algorithm has a finite description. So finally what will be faster: to read """Sum_{n \geq 1} 1/10^n"""" or to catch up the Turtle? This depends only on how long is the distance I let the Turtle in its advantage. 

Dear Dr. Mihai Prunescu , thank you very much for your frank points of view.

Some people believe there is still Ancient Zeno’s Achilles--Turtle Race Paradox in present science system while some do not; I belong to the first group and you belong to the second group. Well, let’s forget Zeno’s Achilles--Turtle Race Paradox and just talk about mathematics.

 Dear Dr. Mihai Prunescu , would you please give your frank points of view to following two questions?

1, can we produce infinite items each bigger than 10000000000000000 (in fact, any constants) from the Un--->0 Harmonic Series” and change the Un--->0 Harmonic Series into a Un--->10000000000000000000 (in fact, any constants) Infinite Series?

2, is Un--->0 = Un--->10000000000000000000 (any constants) in the theory of infinite series in mathematics? What is the meaning of Un--->0 and what is the meaning of Un--->10000000000000000000 (in fact, any constants)?

Yours sincerely,

Geng

Dear Geng, I still do not understand your question (you say  """change the Un--->0 Harmonic Series into a Un--->1000000"""....) 

However there exist a very rich theory of FORMAL SERIES OVER FIELDS. You have series like Sum a_n X^n with coefficients a_n in finite fields, or in Q or in p-adic fields. And there are many interesting questions like if such a series is algebraic or transcendental over the field of rational functions.

Further there is a very nice theory of infinite quantities, containing also various limit constructions of series with infinite terms - the theory of the field """No""" OF ALL NUMBERS by John Norton Conway (see the genial book "Over numbers and games" by Conway). Moreover there are many applications of formal series in Conway's theory.

So I think that if you study those very nice and powerful well defined mathematical instruments you will find answers to your actual questions and also to many other questions you still did not think about. 

Dear Dr. Miguel Ángel Montes and Dr. Mihai Prunescu, thank you.

Well, let’s just focus on one question:

Can we produce infinite items each bigger than10000000000000000000 (in fact, any constants) from the Un--->0 Harmonic Series in human mathematics and how?

Best regards,

Geng

Dear Geng,

Of course we do. This can be done taking sufficiently many terms in harmonic series, summing (computer) and comparing with the given constant (computer). Thus we find a suitable partial sum. Then all the following partial sums have the same property, since the sequence of the partial sums is increasing.

Sincerely, Octav

Dear Dr. Octav Olteanu, thank you.

Could you try to produce first 10000000000000000000  from the Un--->0 Harmonic Series to see how many items will be consume from Harmonic Series?

Best regards,

Geng

Dear Geng,

I would not try to do this. But it does not mean that it is impossible. On the contrary, and this is important: it can be done, eventually using a program for the computer.  

Best regards,

Octav

Dear Geng,

You are confusing with real world problem with computer approximation. It is not possible to produce a huge number using any method in the world. Whenever some one gives a number, by adding 1 to the number, you will get a bigger number.

Could you try to produce first 10000000000000000000 by computer from the Un--->0 Harmonic Series to see how many items will be consume from Harmonic Series?

Yes. We can produce. Take any number N, then sum from 1 to 2^2N, definitely, you will get N+1. 1+1/2+1/3....+1/2^N >= 1+N/2. You can verify this. So, whenever you give an N, I can construct a 2^2N and produce a number bigger than N. Hence it is divergent.

So for your 10000000000000000000, 2^20000000000000000000 will give the requires result. Computer can't think outside the box. But, human can think.

Dear Dr. Octav Olteanu and Dr. Panchatcharam Mariappan, thank you.

It seems so easy for both of you to prove that infinite items each bigger than10000000000000000000 (in fact, any constants) can be produced from the Un--->0 Harmonic Series in present infinite related mathematics system.

But are you still so sure of your proof (not just an empty fantasy) if you can not prove how many items exactly will be consume from Harmonic Series for the first 10000000000000000000 (I think it is not too much requiring for the proof of just the first one of infinite items of10000000000000000000)?

Best regards,

Geng

I am sure. Whenever you give N (does not matter how big it is), I can produce 2^2N, which would produce the harmonic series sum at least 1 bigger than N.

Also, mathematics has more numbers for which you can't write all of its digits, for example, e, pi, 1/3, 1/7 etc. So, if you ask me to write 2^2N for a large N, we can't write that in our life time, but that does not mean that those numbers do not exist.

Dear Panchatcharam,

Because humans think, they can do a lot of inventive and useful things, including the computer and the attached programs. The purpose is to save time and money. Of course, there are many ingenious problems which cannot be solved by a computer. However, its utility is incontestable in our days. Otherwise, the humans would not invented it. 

Best regards,

Octav  

Dear Octav,

I agree that computer saves time and money. Also, it minimized our job in many ways. My work is completely related to computer and mathematics which feeds me everyday. So, my intention is not to insult computer or its work. However, it can't think out of the box. As per the context, Geng says that a computer can't produce a large number 2^2*google, so it won't exist. So, I argued in that way. For example, a computer can't produce rice or potatoes to eat. We need farmers to do that. It does not mean that edible items don't exist.

Dear Dr. Panchatcharam Mariappan, thank you.

I am sorry, you misunderstand.

I never said “that a computer can't produce a large number bigger than 10000000000000000000”, I just said “can you prove (not just an empty fantasy) the exact items (how many) being consumed from Harmonic Series for the first 10000000000000000000 (I think it is not too much requiring for the proof of just the first one of infinite items of 10000000000000000000)”.

Dear Dr. Panchatcharam Mariappan, do you really know (not just an empty fantasy) how many, how?

Thank you again and looking forward to knowing your solution.

Best regards,

Geng

It is 2^20000000000000000000.

Sorry. I thought "Could you try to produce first 10000000000000000000 by computer" means, it is not possible to produce such a number.

That series diverges, but diverges slowly. Because sum_{1 to 2^k}(1/k)>=1+k/2. If you have to move from k to k+1, in the series, you have to move from 2^2k to 2^{2k+2}. The relation between number of terms and the number is 2^2N. That means, for each N, the sum requires an exponential amount of terms (at least 2^2N).

 Dear Dr. Jean-Pierre Magnot  and Dr.Panchatcharam Mariappan, thank you for your frank points of view.

@ Jean-Pierre: according to my studies, in front of the defects disclosed by the suspended infinite related paradox families, non standard analysis can do nothing at all.

@ Panchatcharam:

1, Re: “It is 2^20000000000000000000.”

 I really mean 10000000000000000000---------in fact, any constants

2, Re: “Can you prove (unnecessary just by computer but any way you like) the exact items (how many) being consumed from Harmonic Series for the first 10000000000000000000 (I think it is not too much requiring for the proof of just the first one of infinite items of 10000000000000000000)”

I really mean: it is not possible to produce such a number.

3, If we can not produce just the first 10000000000000000000, how can we produce infinite items of 10000000000000000000?

4, Mathematics is a science but not fantasy; it not only explains “how to act” but also “what it is in the objective and subjective world”.

Thank you again.

Sincerely,

Geng

Let us consider sum_{n=1 to N}(1/n)=s.

Sorry, I don't understand your question.

(1)Do you want the sum value (s) as 10000000000000000000? or

(2) Do you want the sum of first 10000000000000000000 terms (N)?

(1) To get s=10000000000000000000, you need N = 2^20000000000000000000.

(2) For N = 10000000000000000000 = 1e+19, s = 32 (approx)

Also, for N = 1e+301, s=500.

But, still the relation between N and s is as follows, if N = 2^2k, s=1+k. That means, if k->infinity, both N and s tends to infinity, where N goes in exponential rate, whereas s goes in linear way.

So, I leave another basic analysis question for you. Let f(x)=e^x, what is the relation between x and e^x, how do they move when x->infinity? If you could answer this question, you will get an answer for this.

 Dear Dr.Panchatcharam Mariappan, thank you.

There are two questions in my post:

1, Can we produce infinite items each bigger than1/2, or 100, or 1000000, or 10000000000000000000,… (in fact, any constants) from the Un--->0 Harmonic Series?

Your answer is YES.

2, do you know the exact items (how many) being consumed from Harmonic Series for just the first 10000000000000000000 of those infinite items each bigger than 10000000000000000000?

Your answer is also YES.

Now I am asking HOW.

Thank you again and looking for your solutions.

Sincerely,

Geng

That is simple. The answer lies in my post itself. However, I repeat it again

Consider finite harmonic series.

Immediately you can get the following result.

sum_{1 to 2^2k} (1/n) >= 1+(k/2). It is clear without doubts.

https://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29

Dear Dr.Panchatcharam Mariappan, thank you.

I am sorry you didn’t see clearly my 2 question:

1, Can we produce infinite items each bigger than any constants (such as 10000000000000000000) from the Un--->0 Harmonic Series?

Your answer is YES.

2, do you know the exact items (how many) being consumed from Harmonic Series for just the first 10000000000000000000 of those infinite items each bigger than 10000000000000000000?

Your answer is also YES.

Now I am asking HOW.

Thank you again and looking for your solutions.

Sincerely,

Geng

Sorry. I don't understand your question. Could you explain it in mathematical way.

Consider sum_{1 to N} (1/n) = s. Could you explain that in terms of N, n and s?

"Can we produce infinite items each bigger than any constants?"

Yes. We can. Using archimedean property.

Dear Dr.Panchatcharam Mariappan, thank you.

Re: Could you explain it in mathematical way.

I just hope not only you, but many researchers from different fields can also know what we are discussing about.

Re: Yes. We can. Using archimedean property.

I don’t think there is archimedean property for those Un--->0 items in Harmonic Series and I really never care whether harmonic series is divergent or convergent at all. I just focus on:

1, what Un---->0 in Harmonic Series means, what differences between y--->0 (Un---->0) in Harmonic Series and y--->0 (dx ---->0) in calculus, how we treat those “y--->0 things”;

2, why there are the fundamental defects of “infinite theory, limit theory, number theory” disclosed by the suspended infinite related paradox families in present infinite related science theory system, how to solve such defects.

Sincerely,

Geng

after reading the question, I remembered non-Standard  Models too. a fluent book is non standard Analysis written by Gold Blatt

Harmonic series Un->0 is a particular case of calculus Un->0 or x->0, where x belongs to R. I don't think there is a calculus for N, Q, Z. As far as I know, Calculus talks about R and C. Experts should pardon me, if I omit something or said wrong about calculus.

 Dear Dr. Maryam M Ajerlou and Dr. Panchatcharam Mariappan, thank you.

@ Dr. Maryam M Ajerlou: according to my studies, in front of the defects disclosed by the suspended infinite related paradox families, Non Standard Analysis can do nothing at all ------ it can only do what the Standard Analysis can.

@ Dr. Panchatcharam Mariappan, do you think there are “potential infinitesimal” and “actual infinitesimal” in mathematics? y--->0 (Un---->0) in Harmonic Series is a typical “potential infinitesimal” and y--->0 (dx ---->0) in calculus is a typical “actual infinitesimal”, they are different “y--->0 things”--------the cognizing and treating ways to them must be different of cause.

Sincerely,

Geng

I don't know math very much( I'm student ) but as far as I have seen there have been many tools in mathematics for other fields of science, specially in pure math, so I thought your question is connected to a theory in math, certainly. like Hyperreals in non standard Analysis of R ( R^* ),( and of course hyperrationals, hyperintegers,.. ) this is a new definition of real numbers structure that introduced by Robinson. a special attitude to numbers , specially infinite numbers and infinitesimal,  as Dr. Prunescu explained it exactly. Nice part is that  for every hyperreal b  in R^* is assigned halo of b and also galaxy of b, this helps to have a good Intuition to infinite. the point is the concept of limit or ..will be tangible in R^*. /  what is limit is human science? I mean you calculate limit of what object in the theory? you have a sequence of what?  that want to know they converges to a number./ may you tell me a simple example about limit ? /  you told  it can only do what the Standard Analysis can, but I saw something in it, one can prove many theorems shorter and in a better way by it,and  likely it has many others usage that I don't know them.  

Dear Mr. Maryam M Ajerlou , thank you.

The problem is: in front of the defects disclosed by the suspended infinite related paradox families, “many tools”  in present infinite related classical mathematics system (including Non Standard Analysis) can do nothing at all ------ they are just the different formal languages with the same nature being confined to present classical mathematics system with the foundation of “potential infinite” and “actual infinite”.

Regards,

Geng

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

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

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

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

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

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

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

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

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

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

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V