= Second-moment expansion of a Laplace transform
{title2=$\mathbb Ee^{-\lambda X}=1-\lambda\mathbb EX+\lambda^2\mathbb EX^2/2+o(\lambda^2)$}
For nonnegative $X$ with finite <second moment>, <differentiation> under the <expectation> gives $\phi'(\lambda)=-\mathbb E[Xe^{-\lambda X}]$ 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.
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