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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On two conjectures concerning the partial sums of the harmonic series


Author: Stephen M. Zemyan
Journal: Proc. Amer. Math. Soc. 95 (1985), 83-86
MSC: Primary 40A05
MathSciNet review: 796451
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Abstract: Let $ {S_n}$ denote the $ n$th partial sum of the harmonic series. For a given positive integer $ k > 1$, there exists a unique integer $ {n_k}$ such that $ {S_{{n_k} - 1}} < k < {S_{{n_k}}}$. It has been conjectured that $ {n_k}$ is equal to the integer nearest $ {e^{k - y}}$, where $ \gamma $ is Euler's constant. We provide an estimate on $ {n_k}$ which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which $ {S_{{n_k} - 1}}$ and $ {S_{{n_k}}}$ differ from $ k$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0796451-8
PII: S 0002-9939(1985)0796451-8
Article copyright: © Copyright 1985 American Mathematical Society