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

Abstract

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!/n→1/en \root n \of {n!/n \to 1/e} . A generalization of this limit is (11s· 22s· · · nns)1/ns+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.