A further generalization of _n→∞[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! 2πn(n/e)n, that [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.