A further generalization of $\lim_{n\to\infty}\sqrt[n]{n!}/n = 1/e$
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 (11^s · 22^s ··· nn^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 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
limit formula; generalization; prime numbers; perfect powers
References
R.C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. 72 (1996), 261–280.
R.C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II, Proc. London Math. Soc. 83 (2001), 532–562.
F. Burk, S.K. Goel, and D.M. Rodriguez, Using Riemann sums in evaluating a familiar limit, College Math. J. 17 (1986), 170–171.
R. Jakimczuk, Some results on the difference between consecutive perfect powers, Gulf J. Math. 3 (2015), 9–32.
B.J. McCartin, e: The master of all, Math. Intelligencer 28 (2006), 10–21.
J. Rey Pastor, P. Pi Calleja, and C.A. Trejo, Análisis Matemático, Vol. 1, Editorial Kapelusz, Buenos Aires, 1969.
N. Schaumberger, Alternate approaches to two familiar results, College Math. J. 15 (1984), 422–423.
N. Schaumberger, A generalization of $lim_{ntoinfty}sqrt[n]{n!}/n = e^{-1}$, College Math. J. 20 (1989), 416–418.
Department of Statistics, Razi University, Iran Iran, Islamic Republic of
https://orcid.org/0000-0003-4027-9838
División Matemática, Universidad Nacional de Luján, República Argentina Argentina
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