There are two “reasonable” limit operations in convergent- divergent proof of Harmonious Series:
1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）
=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２）
>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3）
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）
（1） During the whole process in dealing with infinite substances (infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried our forever. This situation has been existing in mathematics since antiquity------ those items of Un--->0 never be 0 all the time and Harmonious Series is divergent, so we can produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonious Series and change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity. Here, with limit theory and technique, we see a “strict mathematically proven” modern version of ancient Zeno’s Paradox.
（2） During the process in dealing with infinite substances (infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation has been existing in mathematics since antiquity-------those items of Un--->0 must be 0 from some time and Harmonious Series is not divergent, so we cannot produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…from infinite Un--->0 items in Harmonious Series. But if it is convergent, another paradox appears.
But when and to which should or should not people treat infinitesimals appearing in infinite numeral cognitions that way?
Does limit theory need basic theory, what is it?
You can refer to this web site. It is one of the top websites on Theory of Constraints.
http://www.goldratt.com/
Dear Mr. Ahmad Esmaili,
Thank you. Could I have your idea on my above presentation?
Sincerely yours!
Geng
Dear Mr. Robert Lowen,
A very important feature acknowledged for “non-standard analysis” is its deep structural equivalence with “standard analysis” and this is just because nonstandard structure is constructed within the confines of present classical infinite related theory system ------“standard analysis” can do anything “non-standard analysis” can.
So, in infinitesimal related mathematical area, “non-standard analysis” has the same fate as any former theories and is also unable to solve any problems and defects disclosed by the suspended infinitesimal paradox family at all.
This situation has been studied in my papers uploaded on RG.
Sincerely yours!
Geng
If we understand the foundation and meaning of a certain mathematical operation, we can use it well and express self-justification why we behave that way theoretically and practically; but if we don’t understand the foundation and meaning of a certain mathematical operation, we may meet troubles-------people have been troubling by the “non-number infinitesimal (variables)” related "tyranny of epsilons and deltas limit operation" in standard analysis ever since.
Now, “non-standard analysis” is on its way with its “infinitesimal operation” instead of the " epsilons and deltas limit operation" in standard analysis. A very important feature acknowledged for “non-standard analysis” is its deep structural equivalence with “standard analysis” and this is just because nonstandard structure is constructed within the confines of present classical infinite related theory system-----all the problems and defects disclosed by the suspended infinite paradoxes (such as the newly discovered Harmonious Series Paradox) will be shifted to “non-standard analysis” and people don’t understand the foundation and meaning of the newly invented “being-number infinitesimal” either.
So, will the new “being-number infinitesimal” related “tyranny of infinitesimal operation” in non-standard analysis trouble us from now on?
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
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