### 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 nk such that S_{n}_{k-1}<S_{n}_{k}. It has been conjectured that n_{ki} is equal to the integer nearest e^{k-γ}, where γ 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-} and S_{n}_{k} differ from k.

Original language | English (US) |
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Pages (from-to) | 83-86 |

Number of pages | 4 |

Journal | Proceedings of the American Mathematical Society |

Volume | 95 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1985 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Proceedings of the American Mathematical Society*, vol. 95, no. 1, pp. 83-86. https://doi.org/10.1090/S0002-9939-1985-0796451-8

**On two conjectures concerning the partial sums of the harmonic series.** / Zemyan, Stephen M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On two conjectures concerning the partial sums of the harmonic series

AU - Zemyan, Stephen M.

PY - 1985/9

Y1 - 1985/9

N2 - Let Sn denote the nth partial sum of the harmonic series. For a given positive integer k>1, there exists a unique integer nk such that Snk-1<Snk. It has been conjectured that nki is equal to the integer nearest ek-γ, where γ is Euler's constant. We provide an estimate on nk which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which Snk- and Snk differ from k.

AB - Let Sn denote the nth partial sum of the harmonic series. For a given positive integer k>1, there exists a unique integer nk such that Snk-1<Snk. It has been conjectured that nki is equal to the integer nearest ek-γ, where γ is Euler's constant. We provide an estimate on nk which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which Snk- and Snk differ from k.

UR - http://www.scopus.com/inward/record.url?scp=84907928734&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907928734&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-1985-0796451-8

DO - 10.1090/S0002-9939-1985-0796451-8

M3 - Article

AN - SCOPUS:84907928734

VL - 95

SP - 83

EP - 86

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -