Second-moment expansion of a Laplace transform ($\mathbb{E}{e}^{-\lambda X}=1-\lambda \mathbb{E}X+{\lambda }^2\mathbb{E}{X}^2/2+o({\lambda }^2)$)

For nonnegative $X$ with finite second moment, differentiation under the expectation gives ${\varphi }^{\prime }(\lambda )=-\mathbb{E}[X{e}^{-\lambda X}]$ for $\lambda >0$. The exponential Taylor remainder divided by ${\lambda }^2$ tends pointwise to zero and is bounded in absolute value by $X^2/2$. The dominated convergence theorem therefore proves the displayed expansion without requiring a third moment.