Path: ...!weretis.net!feeder8.news.weretis.net!news.szaf.org!news.karotte.org!news.space.net!news.muc.de!.POSTED.news.muc.de!not-for-mail From: Alan Mackenzie 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. Message-ID: References: MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Injection-Date: Sat, 8 Mar 2025 14:03:48 -0000 (UTC) Injection-Info: news.muc.de; posting-host="news.muc.de:2001:608:1000::2"; logging-data="66054"; mail-complaints-to="news-admin@muc.de" User-Agent: tin/2.6.4-20241224 ("Helmsdale") (FreeBSD/14.2-RELEASE-p1 (amd64)) Bytes: 2958 Lines: 61 WM 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).