A further generalization of $\lim_{n\to\infty}\sqrt[n]{n!}/n = 1/e$

Reza Farhadian; Rafael Jakimczuk

Published: 2022-04-18
Language: EN

Vol. 36 No. 2 (2022)

A further generalization of $\lim_{n\to\infty}\sqrt[n]{n!}/n = 1/e$

Reza Farhadian

, Rafael Jakimczuk

Abstract

It is well-known, as follows from the Stirling's approximation n! ~ \sqrt{2πn}(n/e)ⁿ, that \sqrt[n]{n!}/n→1/e. A generalization of this limit is (1^{1^s} · 2^{2^s} ··· n^{n^s})^{1/n^{s+1}} · n^-1/(s+1)→e^{-1/(s+1)^2} which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger's formula for a general sequence A_n of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.

(EN)

Keywords:

limit formula , generalization , prime numbers , perfect powers (EN)

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Farhadian, R., & Jakimczuk, R. (2022). A further generalization of $\lim_{n\to\infty}\sqrt[n]{n!}/n = 1/e$. Annales Mathematicae Silesianae, 36(2), 167–175. Retrieved from https://trrest.vot.pl/ojsus/index.php/AMSIL/article/view/13549

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Reza Farhadian

farhadian.reza@yahoo.com
https://orcid.org/0000-0003-4027-9838

Affiliation:

Department of Statistics, Razi University, Iran Iran, Islamic Republic of

Rafael Jakimczuk

Affiliation:

División Matemática, Universidad Nacional de Luján, República Argentina Argentina

Vol. 36 No. 2 (2022)
Published: 2022-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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