A further generalization of $\lim_{n→∞}\sqrt[n]{n!}/n = 1/e$ - Farhadian, Reza; Jakimczuk, Rafael (Annales Mathematicae Silesianae, 2022)
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.