Number series and computer

Authors

Petr Eisenmann
Martin Kuril

Volume

16

Issue

1-3

Abstract

This article describes the discovery and the subsequent proof of a hypothesis concerning harmonic series. The whole situation happened directly in the course of the process of a maths lesson. The formulation of the hypothesis was supported by the computer. The hypothesis concerns an interesting connection between harmonic series and the Euler's number 𝑒. Let 𝐻_𝑛 denote the 𝑛’th partial sum of the harmonic series. Let’s notice the sums 𝐻₁, 𝐻₄, 𝐻₁₁, . . ., where the partial sum reaches 1, 2, 3, . . . for the first time. Let’s mark the relevant indexes 1, 4, 11, . . . as 𝑝₁, 𝑝₂, 𝑝₃, . . .. So 𝑝_𝑛 is the index of such a partial sum for which the following is true: 𝐻_P𝑛−1 < n, 𝐻_P𝑛 > n. The hypothesis lim P_n+1/P_n = e has been proved, in an elementary way, in the article.

First Page

253

Last Page

262

Recommended Citation

Eisenmann, Petr and Kuril, Martin (2019) "Number series and computer," The Mathematics Enthusiast: Vol. 16 : No. 1 , Article 12.
DOI: https://doi.org/10.54870/1551-3440.1457
Available at: https://scholarworks.umt.edu/tme/vol16/iss1/12

Digital Object Identifier (DOI)

10.54870/1551-3440.1457