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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: The existence of dark numbers proven by the thinned out harmonic series
Date: Wed, 12 Mar 2025 10:22:01 -0000 (UTC)
Organization: muc.de e.V.
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WM <invalid@no.org> wrote:
> The harmonic series diverges. Kempner has shown in 1914 that when all 
> terms containing the digit 9 are removed, the series converges.

> 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.

That's so slovenlily worded it's hardly meaningful.  Does the adverb
"simultaneously" qualify "containing" or "diverge"?  As I've told you
before, the terms don't diverge; their sum does.

What I think you're trying to say is that the sum of all terms
containing both 8 and 9 diverges.  It is far from clear that this is the
case.

> We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8, 
> 9, 0  in the denominator without changing this.

What do you mean by "this"?  What does it refer to?

> That means that only the terms containing all these digits together
> constitute the diverging series. (*)

There are many diverging sub-series possible.  I think you mean "a"
diverging series.

> But that's not the end! We can remove any number, like 2025, ....

From what?  2025 isn't a digit.

> .... and the remaining series will converge. For proof use base 2026.
> This extends to every definable number.

Meaningless.  "Definable number" is itself undefined.

> Therefore the diverging part of the harmonic series is constituted
> only by terms containing a digit sequence of all definable numbers.

Gibberish.

> Note that here not only the first terms are cut off but that many 
> following terms are excluded from the diverging remainder.

> This is a proof of the huge set of undefinable or dark numbers.

In your dreams.

> (*) At this point the diverging series starts with the smallest term 
> 1023456789 and contains further terms like 1203456789 or 1234567891010 
> or 123456789111 or 1234567891011. Only those containing the digit 
> sequence 10 will survive the next step, and only those containing the 
> digit sequence 1234567891011 (where the order of the first nine digits 
> is irrelevant) will survive the next step.

> Regards, WM

-- 
Alan Mackenzie (Nuremberg, Germany).