  In this note, we show that any continued fraction convergent of the number $e
= 2.71828...$ can be derived by approximating some integral $I_{n, m} :=
\int_{0}^{1} x^n (1 - x)^m e^x d x$ $(n, m \in \mathbb{N})$ by 0. In addition,
we present a new way for finding again the well-known regular continued
fraction expansion of $e$.
