Discussion Overview
The discussion revolves around the effects of deleting specific digits, particularly the digit '9', from the denominators of terms in the harmonic series. Participants explore whether this deletion affects the upper bounds of the series and consider generalizations to other digits. The scope includes mathematical reasoning and exploratory proofs related to series convergence and bounding techniques.
Discussion Character
- Exploratory
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant proposes proving that the sum of the harmonic series is less than 80 if terms with '9' in the denominator are eliminated.
- Another participant questions whether the exclusion of terms like 1/19 is correctly understood as part of the problem.
- Some participants discuss the need to prove that the sum remains bounded above by 80, while others suggest different approaches to bounding the sum.
- A participant suggests counting the terms excluded in each "decade" of numbers to establish an upper bound for the sum.
- There is a proposal to derive a formula for the sum of reciprocals that exclude terms containing '9', but uncertainty remains about the existence of such formulas.
- One participant presents a method to count numbers without the digit '9' across different ranges, leading to a proposed upper bound of 45 for the sum.
- Another participant expresses confusion about whether the discussion proves the upper bound of the original harmonic series sum is 80.
- There is a mention of the divergence of the harmonic series and a correction regarding the count of numbers not containing '9'.
Areas of Agreement / Disagreement
Participants express various viewpoints and methods, with no consensus reached on the validity of the proposed proofs or the correctness of the calculations. Disagreements exist regarding the counting of terms and the implications for the upper bounds of the series.
Contextual Notes
Some participants note the difficulty in establishing exact formulas for the sums involved, leading to reliance on bounding techniques. There are also discussions about the assumptions made in counting terms and the convergence of the series.