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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The truncated harmonic series diverges.
Date: Sat, 8 Mar 2025 15:18:51 +0100
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On 08.03.2025 15:03, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> On 05.03.2025 18:18, efji wrote:
>>> 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.
> 
>> Mistake. That means that the terms containing 9 diverge.
> 
> Mistake.  Terms don't diverge, a series may or may not do so.

A series consists of its terms. It can be expressed briefly as I did or 
clumsy as you prefer.
> 
>>> ???
>>> Kempner has shown in 1914 that the harmonic series CONVERGES if you omit
>>> all terms whose denominator expressed in base 10 contains the digit 9.
> 
>> That means that the terms containing 9 diverge.
> 
> See above.

Learn my brief description.
> 
>> Same is true when all terms containing 8 are removed.
> 
> That remains to be proven, I think.

You are in error. I will show you how the case of 9 works. If you have 
understood, your doubts will turn out groundless. 
https://www.hs-augsburg.de/~mueckenh/HI/HI02 p.15.
> 
>> 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. 

Stop your clumsy waffle.

 > That remains to be proven, though I would guess it is true.

Learn the case of 9, then you will know it.
> 
>> We can continue and remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,
>> 9 in the denominator without changing this. That means that only the
>> terms containing all these digits together constitute the diverging series.
> 
> 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.

No, all the subseries' converge. The remainder diverges.
> 
>> 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.
> 
> For some value of "extends".  I think you're trying to gloss over some
> falsehood, here.
> 
> "Definable" is here undefined and meaningless.

Definable is all that you can think of or communicate as an individual 
number.
> 
>> Therefore the diverging part of the harmonic series is constituted
>> only by terms containing a digit sequence of all definable numbers.

Regards, WM