Quadratic expansion of a cumulant-generating function ($\log M_X(t)=\mu t+\frac{1}{2}{\sigma }^2{t}^2+o(t^2)$)

Suppose the moment-generating function $M_X$ is finite on an open interval around zero. Its cumulant-generating function $K_X=\log M_X$ satisfies $K_X(0)=0$, $K_X^{\prime }(0)=\mathbb{E}X=\mu$ and $K_X^{\prime \prime }(0)=\mathrm{Var}(X)={\sigma }^2$. Indeed,and $M_X(0)=1$. The Taylor expansion gives the displayed formula, with little o notation describing the remainder. Finiteness in a neighborhood justifies differentiation: polynomial factors times $e^{tX}$ are dominated on a smaller interval by constants times two slightly larger exponential moments.