Convergence rate of the moment generating function

Question

Let $S_n\sim Bin(n,p)$. I would like to obtain an upper bound of \begin{align} \left|E\left[e^{h\frac{S_n-np}{\sqrt{np(1-p)}}}\right]-e^{h^2/2}\right| \ (1) \end{align} (using $n,h,p$). In my understanding, $E\left[e^{h\frac{S_n-np}{\sqrt{np(1-p)}}}\right]=e^{-h\frac{np}{\sqrt{np(1-p)}}}\left(1-p+pe^{\frac{h}{\sqrt{np(1-p)}}}\right)^n$ converges to $e^{h^2/2}$ as $n\to \infty$ from the central limit theorem.

Answer 1

$$\begin{align} \Delta=\left|E\left[e^{h\frac{S_n-np}{\sqrt{np(1-p)}}}\right]-e^{\frac{h^2}{2}}\right| \end{align}$$

Continue the expansion using first logarithms and then Taylor series for large values of $n$ to make $$E\left[e^{h\frac{S_n-np}{\sqrt{np(1-p)}}}\right]=e^{\frac{h^2}{2}}+h^3\,e^{\frac{h^2}{2}}\frac{(1-2 p)}{6 \sqrt{(1-p) p}}\frac{1}{\sqrt{n}}+O\left(\frac{1}{n}\right)$$