Re: The truncated harmonic series diverges.

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