Re: The existence of dark numbers proven by the thinned out harmonic series

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).