What is the relation between x and e^x, if x->infinity e^x->infinity. Are they two different infinities, like potential and actual?

Dear Dr.Panchatcharam Mariappan, thank you for the smart question.

Yes!

There are “potential infinite” and “actual infinite” in our present infinite related classical science theory system, so we can not avoid (logically have) “potential infinitesimal” and “actual infinitesimal”, “potential infinitebig” and “actual infinitebig” in mathematics.

I think there should be such questions 2500 years earlier.

Sincerely,

Geng

Ok, now could you answer the following, is x->infinity potential or actual? is e^x->infinity potential or actual?

You are ultimately trying to prove that the harmonic series is convergent. Am I right? But it diverges, please refer this proof.

https://web.williams.edu/Mathematics/lg5/harmonic.pdf

Dear Dr.Panchatcharam Mariappan, thank you for the smart questions.

@ The first post: At present, I am sorry to say it is not easy to answer your question:

1, the endless debate on “what is potential infinite and how it being” and “what is actual infinite and how it being” is still on now and they are still very confusing; are you sure you are not confused on them (may be we two have different views too)?

2, the same as “is Un--->0 potential infinitesimal or actual infinitesimal?”, the question in your post “is x->infinity potential or actual? is e^x->infinity potential or actual?” depends on where they are.

@ The second post: exactly NO.

I just care those defects through Harmonic Series Paradox in the foundation of “infinite theory, limit theory, number theory” in present infinite related science theory system (convergence or divergence of Harmonic Series is nothing important in the whole new working field).

1, no one cares what Un--->0 in Harmonic Series means-------“potential infinitesimal” or “actual infinitesimal”,

2, no one cares what differences between y--->0 (Un---->0) in Harmonic Series and y--->0 (dx ---->0)  in calculus,

3, no one cares whether or not we can really produce (with infinite theory) infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000000000000,… from the Un--->0 Harmonic Series and no one cares the exact (how many) items being consumed for just the first 10000000000000000000 from Harmonic Series.

Sincerely,

Geng

I am sure. The series diverges. But, I got another question in mind.

As per the bracket-placing rule, sum_{n=1 to 2^2k} (1/n)>=1+k. So, that means, it requires at least 2^2k terms to get k+1. If k -> infinity, it is sure both 2^2k and the harmonic series divergent. But I have the following question, for which I don't have an answer.

Let X={1,2,...n}, P(X)=2^X. Then if n -> infinity, will |X| = aleph-null and |P(X)|=2^{aleph-null}=cardinality of continuum?

If this statement is true, I am afraid that the series requires uncountable numbers.

Dear Dr.Panchatcharam Mariappan, thank you

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

Dear Geng

 do you mean by Un  in harmonic series the general term of  harmonic series ?

if so then where is the problem?

the same sequnce(Un) goes to zero

if no then please explain( at least by example)

Regards

What is the problem of reaching Un->0 in calculus and harmonic series. You have to think in the given set. For the harmonic series, Un lies in Q, whereas in calculus it lies in R. You can construct infinite number of sequences which converges to 0 both in Q as well a R.

Dear Dr. Ramadan Sabra and Dr. Panchatcharam Mariappan, thank you.

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”.

 A lot of troubles come to being when “potential infinitesimal” and “actual infinitesimal” are confused.

Sincerely,

Geng

About 3: Maybe no one cares how large N has to be in order for "sum over 1/n from 1 to N" to be larger than, say, C, but it is certainly possible to calculate it. By a simple comparison argument, one can say that the mentioned sum is larger than the integral from 1 to N+1 over 1/x, which can be explicitly evaluated to log(N+1). Hence, we need e^C -1 terms to assure that the partial sum larger than C. Feel free to put C=1000000000 or any other number.

Dear Dr. Axel Flinth, thank you. I had concerned following question before:

How many times can we have the integral operations in the Un--->0 Harmonic Series to get infinite items each bigger than, say, C (for example C=1000000000 or any other number)?

Regards,

Geng

Dear Geng,

Could you tell me please, what does it mean "Un--->0 in Harmonic Series"?

Sincerely, 

Aslanbek

Dear Dr. Aslanbek Naziev, thank you.

I think we can have a clear definition (description) of "Un--->0", "Harmonic Series" and "Un--->0 in Harmonic Series" from mathematics books.

We focus on the differences, in our present infinite related classical mathematics theory system, between y--->0 (Un---->0) in Harmonic Series (potential infinitesimal) and y--->0 (dx ---->0) in calculus (actual infinitesimal).

Sincerely,

Geng

Could you please write the exact definition of Un-->0 in harmonic series and Un-->0 in calculus?

Dear Geng, thank you.

Nevertheless, could you tell (or repeat) me, please  this clear definition (description). It is necessary for we will be sure that we are talking about the same thing.

Sincerely,

Aslanbek 

Dear Dr. Panchatcharam Mariappan and Dr. Aslanbek Naziev, thank you. I am sorry that I didn’t present clearly in previous posts and I have corrected the vague expression.

1, what we are talking about is in the frame of present infinite related classical mathematics theory system.

2, y---->0 (Un---->0 or dx ---->0 or…) tells us two things;

(1), the “being” state (especially as a “being” tendency decided by existing law, existing nature under the definition of “infinite”) of a kind of infinite related things being cognized

(2), the numerical meaning of a kind of infinite related things being cognized

3, the symbol of y---->0 appears in harmonic series (Un---->0) and in calculus (dx ---->0); it has been called infinitesimal long ago.

Sincerely,

Geng

I think you are confusing with limit of a sequence and limit of a function.

Limit of a sequence can be obtained from limit of a function. Because, sequence is a function f, from N, a subset of R, to X. If we explain it in calculus or real analysis way, it is same as stating that limit of a sequence is limit of the function f.

That is limx->y f(x) = L. For harmonic sequence, you can define a function f:R->R and say f(x)=1/x. Now you can define, f|N:R->R by f(n)=1/n. So, both f and f|N approaches the same limit when x->infinity. 

I am talking about a kind of infinite related being--------- “y---->0”.

Restriction on the domain of calculus function would give the sequence, therefore no confusions. Please read more about real analysis and functional analysis. There are many more concepts already developed such as sequence of functions, limit of limit etc. Please read analysis I and analysis II by Terrence Tao and compactness and contradiction by the same author. I hope your doubts would be clear by then.

Dear Dr. Panchatcharam Mariappan, thank you.

I think the definition of infinitesimal is associated with the definition of infinite in science (mathematics):

1, Can we really have many different definitions with different natures for the concept of “infinite” in human in science (mathematics)?

If the answer is yes, then we can have many (how many? as many as we wish?) different definitions with different natures for infinitesimal in science (mathematics).

2, Can we have both “potential infinite” and “actual infinite” in science (mathematics)?

If the answer is yes, then we can have “potential infinitesimal”, “actual infinitesimal” in science (mathematics).

Of cause different people can have (choose) different understandings freely on infinitesimal (it may be expressed as “function” or “non-number variable” by many people) which are linking to the very fundamental areas in the scientific research work for different fruits-------many people think we can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000000000000,… from the Un--->0 Harmonic Series but I don’t.

Sincerely,

Geng

Dear Geng,

1. The answer is No. It is a unique definition and well-defined.

2. Infinity is an abstract concept describing something without any bound.

So infinity is a symbol to denote unboundedness. The number of terms in the sequence (Un) is unbounded. Now, it is enough to prove that harmonic series is unbounded, which is quite easy from the definition of unbounded.

Dear Dr. Panchatcharam Mariappan, thank you.

1, For the first question, I agree that we can have one and only one definition for “infinite”. But, quite a lot people argue that we can really have many different definitions with different natures for the concept of “infinite” in human in science (mathematics).

2, For the second question, quite a lot people argue that we can really have “potential infinite” and “actual infinite” in science (mathematics)------not only infinite. Do you refuse “potential infinite” or “actual infinite”, do you refuse both or what and what to do?

These two questions have been troubling human for more than 2500 years.

Regards,

Geng

Dear Dr. Miguel Ángel Montes, thank you.

Just one question： how many such psychological concepts (as many as we wish?) can we have when we deals with infinity in mathematics?

Best regards,

Geng

If you think potential infinity and actual infinity are two mathematical concepts, could you provide the proper definition for potential infinity and actual infinity? Before discussing about a concept on mathematics, the term should be well defined. Otherwise, there will be more contradictory results.

Dear Dr. Panchatcharam Mariappan, thank you.

The very trouble is: people accept “potential infinity” and “actual infinity”, but no one so far in the world can provide the proper definition for potential infinity and actual infinity and people have been keeping arguing and debating on the natures of them ever since-------endless and unsolvable.

This is one of the fatal defects in present infinite related classical mathematics system： no one knows what is in front of you when treating “infinite things”--------“potential infinity” or “actual infinity”, how and what to do with different infinity?

Best regards,

Geng

So, you accepted that it is not a well defined term. Then why we should blame no one cares about it. Some of the terms can't be defined properly, for example, beauty, taller, shorter, longer etc. Since these terms are not well defined in Mathematics, it does not mean that mathematics itself is wrong or fatal defects available in infinite related classical mathematical system. However, infinity a symbol for unboundedness is well defined, where the unbounded is an adjective for many objects such as unbounded set, unbounded number, unbounded series, unbounded sequence etc. All these terms are well defined. In other words, when we define boundedness, its contrary or whichever is not bounded becomes unbounded.

Further, if you want to define something in between bounded and unbounded, you can get the help of fuzzy logic and you can define bounded as 0 and unbounded as 1 and you can define (????) anything in between as 0.5 bounded, 0.75 bounded, etc. But I can't give any guarantee that there will be well defined term.

I can't guarantee because: Probability theory and fuzzy logic are related to measure theory, which is part of calculus and analysis which contains bounded and unbounded as basic principles. So, by this mean, we are trying to change the basic principle of a concept using a theory developed on already existing basic principles, which looks like an irony.

Conclusion: Since both potential infinity and actual infinity are not well defined, but infinity is well defined, it is better to stick with infinity.

Dear Dr. Panchatcharam Mariappan, thank you.

The “2500 years old huge black cloud of infinite related paradoxes over mathematics sky” is really not easy to be dispelled.

Best regards,

Geng

 It is not necessary that whatever our ancestors believed or stated should be correct. As I have mentioned earlier, concepts which are undiscovered or unexplained 2500 years ago need not be undiscovered or unexplained forever. There are some concepts developed in earlier century and later denied due to wrong assumptions or for not well-defined definitions. For more details, please read History of Mathematics. I state a single point here.

Lagrange, in the late 18th century, believed any function could be expanded into a power series except at some isolated points and wrote an entire book on analysis based on this assumption. (This was a time when there wasn't a modern definition of function; it was just a "formula".) His goal was to develop analysis without using infinitesmals or limits. This approach to analysis was influential for quite a few years. See Section 4.7 of Jahnke's "A History of Analysis". Work in the 19th century, e.g., Dirichlet's better definition of function, blew the whole work of Lagrange apart, although in a reverse historical sense Lagrange was saved since the title of his book is "Theory of Analytic Functions..."

More are listed in this webpage

http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong

Dear Dr. Panchatcharam Mariappan, thank you.

Human science history has really proved a truth for us researchers to lay in mind: HUMAN SCIENCE CAN NOT EXIST WITHOUT METABOLISM.

Best regards,

Geng

Both exist, as an example for potencial infinitesimal

1. See Infinitesimal generators.... in http://www.ncbi.nlm.nih.gov/pmc/articles/PMC432795/

2. And in the book of Jhon L. Bell pp 74 for actual infinitesimal

https://books.google.com.co/books?id=Eq8ZualfMRkC&pg=PA174&lpg=PA174&dq=actual+infinitesimal&source=bl&ots=Nip4EKzD2m&sig=2zGVYbKzOSFuHxdFCcW9S-dLC5g&hl=es&sa=X&ved=0ahUKEwiz9ZKfs7nKAhUELB4KHe-iBEoQ6AEILTAC#v=onepage&q=actual%20infinitesimal&f=false. .

William

Dear Dr. Jose William Porras, thank you very much for your offers.

But it has been a very troublesome problem ever since: how to decide what is in front of you, potential infinitesimal or actual infinitesimal and how to treat them (with the same way or different way)?

Such as:

1, are those y---->0 (Un---->0) items in Harmonic Series potential infinitesimal or actual infinitesimal?

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

Best regards,

Geng

Geng, If both exist in mathematics each one is treated according to the circumstances, that is why I have indicated some references, .... to clarify that......

Best regards, William

Dear Dr. Jose William Porras, thank you.

I agree with you that ”potential infinitesimal and actual infinitesimal should be treated according to the circumstances”. So, there should be some theories and techniques to treat them in different circumstances:

1, are those y---->0 (Un---->0) items in Harmonic Series potential infinitesimal or actual infinitesimal?

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

Since the “theories and techniques to treat them in different circumstances” are still vacant in present infinite related classical mathematics theory system, the above two suspended questions remind us: a totally new working field is waiting to be opened up?!

Best regards,

Geng

Geng, 

I think first it would define its scientific and mathematics applications, before opening a field of research or "a totally new working field is waiting to be opened up?! as you say.

Best regards

William

Dear Dr. Jose William Porras, thank you.

"A totally new working field is waiting to be opened up” seems very hard and long at present.

How do you think of following suspended question:

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

Best regards,

Geng

Dear Geng,

If you can come up with the upper bound for the harmonic series, we will answer your suspended question.

Dear Dr. Panchatcharam Mariappan, thank you.

OK, you are sure?!

One of the upper bounds can be “infinite items each bigger than 10000000000000000000”.

Best regards,Add your answer

Geng

Can be or exactly? What is the meaning of infinite items each bigger than 10000000000000000000? Do you mean that the number 10000000000000000000 is the least upper bound?

If so, please do the harmonic sum for the first 2^{20000000000000000000} terms, definitely you will get a number greater than 10000000000000000001. It is for sure, because the harmonic sum for the first 2^{2k} terms is a value greater than k+1.

Dear Dr. Panchatcharam Mariappan, thank you.

Exactly? It seems we are now playing an interesting game?!

Would you please tell how many items each bigger than 10000000000000000000 can you produce from Harmonic Series------one or two or three or a great many or infinite?

Best regards,

Geng

My question is: What is the least upper bound? Could you produce that? Whenever you say "N is the least upper bound", I can give the harmonic sum with first 2^{2N} terms, which would definitely produce one greater than what you have given (N). So, you can't give the least upper bound. This is the basic principle of unboundedness.

How many: That is also easy to find. Let S1 = sum_{1 to 2^2k} (1/n) and S2 = sum{1 to 2^2m} (1/n). Assume m > k, then S2 - S1 = m - k. So, if you can find m and n such that m-n>10000000000000000000, then you require  2^2m - 2^2k terms. Do you say such m and are impossible to find?

Dear Dr. Panchatcharam Mariappan, thank you.

Well, from the very beginning, my question is: how many items each bigger than 10000000000000000000 (any constants) can be produced from Harmonic Series. It is nothing to do with “the least upper bound”.

Best regards,

Geng

That is also easy to find. Let S1 = sum_{1 to 2^2k} (1/n) and S2 = sum{1 to 2^2m} (1/n). Assume m > k, then S2 - S1 >= m - k. So, if you can find m and k such that m-k>10000000000000000000, then you require 2^2m - 2^2k terms. Do you say such m and k are impossible to find?

It is nothing to do with “the least upper bound”. This looks funny. S={Sk : harmonic sum of first k terms}, where k = 1 to infinity is a set. If you claim S is a bounded set, it should have the least upper bound. 

Dear Dr. Panchatcharam Mariappan, thank you.

I never clam that Harmonic Series is convergent.

Best regards,

Geng

As per the real analysis and calculus, if it is not convergent, then it is divergent.

Dear Dr. Panchatcharam Mariappan, thank you.

So the situation is:

1, I never clam that Harmonic Series is convergent.

2, my question is: how many items each bigger than 10000000000000000000 (any constants) can be produced from Harmonic Series.

Best regards,

Geng

1. The series is not convergent. So divergent.

2. Please see my earlier post. I have given the exact number. 2^{2m} - 2^{2k} items where m-k> any constant you give.

Dear Dr. Panchatcharam Mariappan, thank you.

So the situation is: in present infinite related classical mathematics theory system, (1) no one can prove Harmonic Series is convergent and (2) no one can prove we can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000000000000,…(any constants) from the Un--->0 Harmonic Series.

Either convergent or divergent is wrong (un-scientific) -------- Harmonic Series Paradox, a modern version of Zeno’s paradox family members.

Best regards,

Geng

No. You are wrong. This is my final point. I have already provided enough proof. Please read it properly. I have already mentioned and given proper answer to show that harmonic series is unbounded. Still if you believe it is bounded or not divergent, we cant help anymore. It is your misunderstanding of the mathematical concepts.

Dear Dr. Panchatcharam Mariappan, thank you.

We understand clearly from our science history that at least from Zeno’s time 2500 years ago, we human have been trapped deeply in an irresistible vortex of “confusing abstract concept of infinite and its carriers”-------the confusion of “ontology--form”, the confusion of “abstract concept of fruit and its carriers of banana, apple, grape, …”. This situation produces great ill effects on our cognizing activities to “infinite things”. First, people are unable to know what kind of infinite things are in front of us and how to cognize and treat them scientifically and exactly because we don’t know whether they are “abstract concept of infinite” or “carriers of abstract concept of infinite” or both, “potential infinite things” or “actual infinite things” or both. Second, there are really no theories of “the carriers of potential or actual infinite and their relating treating techniques” in present classical infinite related science and mathematics theory systems.

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