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Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM <wolfgang.mueckenheim@tha.de> Newsgroups: sci.math Subject: The truncated harmonic series diverges. Date: Wed, 5 Mar 2025 11:01:16 +0100 Organization: A noiseless patient Spider Lines: 16 Message-ID: <vq97dc$2bkel$2@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Wed, 05 Mar 2025 11:01:17 +0100 (CET) Injection-Info: dont-email.me; posting-host="f0ac73555f2d7ce3058d3da21f162f89"; logging-data="2478549"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+3F5IteuXA1q9FQSGy8ZrdTJZBvaSMU0g=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:k+d47h2N5H/3Q7WrCLrZ5P6x+oU= Content-Language: en-US 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