The problem disclosed by Zeno’s Paradox is still there and the exactly same idea is still working well. Let’s see one of the modern versions of Zeno’s Paradox
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）
Such an antique proof (given by Oresme in about 1360), though very elementary, can still be found in many current higher mathematical books written in all kinds of languages.
Here, with limit theory and technique, we see a “strict mathematically proven” modern version of ancient Zeno’s Paradox:
1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 100000000000000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant (such as Un’ >10000000000000000000000000) ;
2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 100000000000000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox. So, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it.
Lacking the systematic cognition to “infinitesimal”, no one in the world now can answer following question scientifically and this is the very reason for many “suspended infinite related paradox families” in present classical infinite related mathematics:
Are “dx--->0 infinitesimal” in calculus and “Un--->0 infinitesimal” in Harmonic Series the same things? If ”yes”, why we have totally different operations on them? If ”no”, what are the differences and how to treat them differently and why?
--------Could anyone tells how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the first Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the second Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the third Un’ >10000000000000000000000000?
Zeno s paradox is concerned with the geometric series not the harmonic one. The paradox there is that the series converges. I do not see any paradox with Oresme result as the Harmonic series diverges. There are paradoxes related to divergent series but not the computation in the post.
I am sorry that I didn’t express clearly. Zeno created several paradoxes. The Zeno’s paradox I talk about is “Achilles and tortoise” but not those geometric ones.
Actually our operation in Harmonic Series Paradox is: by producing infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000… from Harmonious Series to change an infinitely decreasing Harmonic 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--------this means that not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise in Zeno’s Paradox, the runner will never catch up with it.
Here, with limit theory and technique, we see a “strict proven” modern version of ancient Zeno’s Paradox.
“Achilles and Tortoise” Zeno’s Paradox has disclosed the "real infinite--potential infinite" fundamental defects in present traditional infinite related science theory system. In this theory system, the critical defects are unsolvable because we are not allowed to forget either "real infinite" or "potential infinite" in such paradox (exactly same situation happen in Harmonic Series Paradox).
That is why this paradox family has been troubling us human for more than 2500 years.
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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