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Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
access type Open Access

A Further Generalization of limnn!/nn=1/e {\lim _{n \to \infty }}\root n \of {n!/n} = 1/e

Published Online: 18 Apr 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 08 Jul 2021
Accepted: 22 Mar 2022
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
Abstract

It is well-known, as follows from the Stirling’s approximation n!2πn(n/e)n n! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}} , that n!/n1/en \root n \of {n!/n \to 1/e} . A generalization of this limit is (11s· 22s· · · nns)1/ns+1 · n1/(s+1) → e1/(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 An 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.

Keywords

MSC 2010

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