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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Dark numbers
Date: Tue, 8 Apr 2025 20:09:48 +0200
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The harmonic series diverges. Kempner has shown in 1914 that when all 
terms containing the digit 9 are removed, the serie converges. Here is a 
simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.

That means that the terms containing 9 diverge. Same is true when all 
terms containing 8 are removed. That means all terms containing 8 and 9 
simultaneously diverge.

We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8, 
9, 0 in the denominator without changing this. That means that only the 
terms containing all these digits together constitute the diverging series.

But that's not the end! We can remove any number, like 2025, and the 
remaining series will converge. For proof use base 2026. This extends to 
every definable number. Therefore the diverging part of the harmonic 
series is constituted only by terms containing a digit sequence of all 
definable numbers.

The terms are tiny but that part of the series diverges. This is a proof 
of the huge set of undefinable or dark numbers.

Regards, WM