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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: The truncated harmonic series diverges.
Date: Sat, 8 Mar 2025 14:03:48 -0000 (UTC)
Organization: muc.de e.V.
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WM <wolfgang.mueckenheim@tha.de> wrote:
> On 05.03.2025 18:18, efji wrote:
>> Le 05/03/2025 =C3=A0 11:01, WM a =C3=A9crit=C2=A0:
>>> The harmonic series diverges. Kempner has shown in 1914 that all term=
s=20
>>> containing the digit 9 can be removed without changing the divergence=
..

> Mistake. That means that the terms containing 9 diverge.

Mistake.  Terms don't diverge, a series may or may not do so.

>> ???
>> Kempner has shown in 1914 that the harmonic series CONVERGES if you om=
it=20
>> all terms whose denominator expressed in base 10 contains the digit 9.

> That means that the terms containing 9 diverge.

See above.

> Same is true when all terms containing 8 are removed.

That remains to be proven, I think.

> That means all terms containing 8 and 9 simultaneously diverge.

That's gibberish.  "That means" is false.  What you're trying to say, I
think, is that the sub-series of the harmonic series formed from terms
whose denominator contain both 8 and 9 in their decimal representation
diverges.  That remains to be proven, though I would guess it is true.

> We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,=
=20
> 9 in the denominator without changing this. That means that only the=20
> terms containing all these digits together constitute the diverging ser=
ies.

It means nothing of the kind.  There is no "the" diverging series in the
sense you mean.  There are many sub-series of the harmonic series which
diverge.

> But that's not the end! We can remove any number, like 2025, and the=20
> remaining series will converge. For proof use base 2026. This extends t=
o=20
> every definable number.

For some value of "extends".  I think you're trying to gloss over some
falsehood, here.

"Definable" is here undefined and meaningless.

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

More gibberish.

> Regards, WM

--=20
Alan Mackenzie (Nuremberg, Germany).