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  • The truncated harmonic series diverges.

    From WM@21:1/5 to All on Wed Mar 5 11:01:16 2025
    The harmonic series diverges. Kempner has shown in 1914 that all terms containing the digit 9 can be removed without changing the divergence.
    Here is a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.

    I think that we can remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,
    9, 0 in the denominator without changing the divergence.

    Further I think we can remove every denominator containig any given
    number like 2025 without changing the divergence.

    Further I think that we can remove the chain of all definable numbers
    without changing the divergence.

    This is a proof of the huge set of dark numbers.

    Regards, WM

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  • From Python@21:1/5 to All on Wed Mar 5 14:26:20 2025
    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.
    Here is a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.

    I think that we can remove all terms containing 1, 2, 3, 4, 5, 6, 7, 8,
    9, 0 in the denominator without changing the divergence.

    Further I think we can remove every denominator containig any given
    number like 2025 without changing the divergence.

    Further I think that we can remove the chain of all definable numbers
    without changing the divergence.

    This is a proof of the huge set of dark numbers.

    Regards, WM

    The sum 0 + 2 + 3 + 4 + 5 > 0

    One can remove all non-null even terms, the sum stays positive : 0 + 3 + 5
    0
    One can remove all non-null odd terms, the sum stays positive : 0 + 2 + 4
    0
    All non-null natural numbers are either odd of even

    "Further" one can "WM-think" that all of them can then be remove, the sum
    will stay positive.

    You, crank Wolfgang Mückenheim, from Hochschule Augsburg, just "proved"
    that 0 > 0.

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  • From WM@21:1/5 to All on Wed Mar 5 19:14:58 2025
    Am 05.03.2025 um 18:18 schrieb efji:
    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.

    Am 05.03.2025 um 11:01 schrieb WM:
    The harmonic series diverges. Kempner has shown in 1914 that all terms containing the digit 9 can be removed without changing the divergence.

    Sorry, this is wrong and has been deleted. Kempner has shown in 1914
    that when all terms containing the digit 9 are removed the series is converging.> Here is a simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p.
    15.
    Therefore all terms containing the digit 9 are diverging.

    Regards, WM

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  • From efji@21:1/5 to All on Wed Mar 5 18:18:04 2025
    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.

    ???
    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.

    https://en.wikipedia.org/wiki/Kempner_series
    --
    F.J.

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  • From WM@21:1/5 to All on Thu Mar 6 07:51:02 2025
    The harmonic series diverges. Kempner has shown in 1914 that when all
    terms containing the digit 9 are removed, the serie converges. Here is a
    simple derivation: https://www.hs-augsburg.de/~mueckenh/HI/ p. 15.

    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.

    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.

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

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

    Regards, WM

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  • From WM@21:1/5 to Python on Thu Mar 6 19:08:00 2025
    On 05.03.2025 15:26, Python wrote:

    The sum 0 + 2 + 3 + 4 + 5 > 0

    One can remove all non-null even terms, the sum stays positive : 0 + 3 + 5
    0
    One can remove all non-null odd terms, the sum stays positive : 0 + 2 + 4
    0
    All non-null natural numbers are either odd of even

    "Further" one can "WM-think" that all of them can then be remove, the
    sum will stay positive.

    You have not understood anything, have you?
    But I made a mistake in the first post. That may be your excuse. I am
    sure however, that you won't understand the correct proof either: The
    truncated harmonic series diverges.

    Regards, WM

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  • From WM@21:1/5 to efji on Sat Mar 8 09:37:14 2025
    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.

    ???
    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. Same is true when all
    terms containing 8 are removed. That means all terms containing 8 and 9 simultaneously diverge.

    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.

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

    Regards, WM

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  • From Alan Mackenzie@21:1/5 to wolfgang.mueckenheim@tha.de on Sat Mar 8 14:03:48 2025
    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.

    ???
    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.

    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,
    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.

    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.

    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

    --
    Alan Mackenzie (Nuremberg, Germany).

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  • From WM@21:1/5 to Alan Mackenzie on Sat Mar 8 15:18:51 2025
    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

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