Dear Geng

I do not like limit theory from a logical point of view. However it provides a quite intuitive way of trying to reach a derivative say as a slope.

Infinitesimals are supposed to be smaller than any predefined small number, but the smallest posible positive real number does not exist. The final step of taking one to zero is never taken.

A game whoes end result is never reached.

You can question it but expecting the mathematical community to move on it is quite another matter.

Regards, juan

Dear Mr. Juan Weisz，

Thank you for your frank idea and I agree with you.

1, limit theory is really a good tool for treating the infinite things in mathematics, the problem is it is misused unavoidable very often because of lacking foundation. The newly discovered Harmonic Series Paradox is a typical example:

Can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by brackets-placing rule with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)?

2, sooner or later, the mathematical community has to do something to move on and get rid of the defects discovered in human science and mathematics.

Yours, 

Geng Ouyang

You might try the purely algebra approach

Take the ring of dual numbers in the form x+ey, where ee=0

Then you can show

f(x+ey)=f(x) + ey( df/dx)

One way to show is from the Taylor expansión, to chop it off

after the second term. Transform y into ey. Another way is to

show it is true for all powers of x, positive or negative.

However it means to go out of the field of real numbers, into the hypercomplex and rings.

Then define derivative from this property, and prove all properties of the derivative.

Dear Mr. Juan Weisz，Thank you for your good idea.

But the problem is: how can we treat those Un--->0 items in Harmonic Series with the purely algebra approach, and what result would that be out?

Yours, 

Geng Ouyang

Geng,

The former method is in principle only for the derivative.

The integral is then just the antiderivative.

For series I only know the standard stuff. ie the Harmonic series

diverges even though its terms are decreasing. As an estimate for a series you can always to an integral, in this case over dx/x and this also diverges.

best regards, juan

Dear Mr. Juan Weisz，one question: do you really agree that we are able to produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by brackets-placing rule with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)?

Yours, 

Geng Ouyang

Yes, there are proofs of divergence relying on such regrouping.

Maybe not in increasing terms but in constant terms.

You have to look around in the Calculus books

It really does diverge.

Can anyone prove how many Un--->0 items in Harmonic Series we use to produce the first three (just the first three!) positive constants of 100000000000000000000000000000--------how many Un--->0 items for the first 100000000000000000000000000000, how many Un--->0 items for the second 100000000000000000000000000000, how many Un--->0 items for the third 100000000000000000000000000000?

First change the series into one that will smaller term by term.

So if it diverges you can be sure that the original series also diverges.

Change the denominator 3 into 4, the denominators 5, 6, 7 into 8,

the denominators 9 through 15 into 16, and so on, so you are summing terms

1/2+1/2+......which must diverge. The number of terms summed

go as 2, 4, 8, ..... This would not be the only way to do it.

Hope this finally clears up doubts

Dear Mr. Juan Weisz，you are talking about the divergent proof of Harmonic Series given by Oresme in about 1360, which can still be found in many current higher mathematical books written in all kinds of languages.

So nowadays, many people agree that we are really able to produce infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000000000000000,…by the brackets-placing rule with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000).

I wish someone is able to Can anyone prove how many Un--->0 items in Harmonic Series we use to produce the first three (just the first three!) positive constants of 100000000000000000000000000000----------how many Un--->0 items for the first 100000000000000000000000000000, how many Un--->0 items for the second 100000000000000000000000000000, how many Un--->0 items for the third 100000000000000000000000000000?

If we are unable to get just three 100000000000000000000000000000,

how can we get infinite 100000000000000000000000000000 from Un--->0 items in Harmonic Series -------not for mathematics but for fun or joking?

Thank you again!

Yours sincerely,

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)

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)