A Note on the Lalescu Sequence

TL;DR

The paper analyzes the Lalescu sequence $a_n = \sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$ and proves it is monotonically decreasing, which implies convergence. It reformulates the problem via $\ell_n = \sqrt[n]{n!}$ and $x_n = \log(\ell_{n+1}/\ell_n)$ and employs an enhanced Stirling expansion together with Robbins bounds to obtain precise asymptotics and critical inequalities. By analyzing $2 \exp\big(\frac{x_{n+1}-x_n}{2}\big) \cosh\big(\frac{x_{n+1}+x_n}{2}\big)$ and showing it is strictly less than $2$ for large $n$, the authors establish eventual monotonicity, then complete the proof with rigorous bounds and a computer-assisted verification for small $n$, yielding full monotonicity. The approach combines higher-order factorial asymptotics with careful infinitesimal-order control and applies to related Gamma-function–based sequences, clarifying the limit value $\lim_{n\to\infty} a_n = 1/e$.

Abstract

We prove that the Lalescu sequence is monotonically decreasing.