Re: The truncated harmonic series diverges.

Le 05/03/2025 à 11:01, WM a écrit :

The harmonic series diverges. Kempner has shown in 1914 that all terms containing the digit 9 can be removed without changing the divergence.
Here is a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.
 I think that we can remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 in the denominator without changing the divergence.
 Further I think we can remove every denominator containig any given number like 2025 without changing the divergence.
 Further I think that we can remove the chain of all definable numbers without changing the divergence.
 This is a proof of the huge set of dark numbers.
 Regards, WM

The sum 0 + 2 + 3 + 4 + 5 > 0
One can remove all non-null even terms, the sum stays positive : 0 + 3 + 5

0

One can remove all non-null odd terms, the sum stays positive : 0 + 2 + 4

0

All non-null natural numbers are either odd of even
"Further" one can "WM-think" that all of them can then be remove, the sum will stay positive.
You, crank Wolfgang Mückenheim, from Hochschule Augsburg, just "proved" that 0 > 0.