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It is well-known, as follows from the Stirling’s approximation  n ! ∼ 2 π n ( n / e ) n n! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}} , that  n ! / n → 1 / e n \root n \of {n!/n \to 1/e} . A generalization of this limit is (1 1 s · 2 2 s · · ·  n n 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  A n  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.

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