To prove CLT of binomial distribution,
$X \sim \mbox{bin}(n,p)$ $M_X(t)=(p e^t+q)^n$ where $M$ is mgf.
Let $Z=\frac{X-np}{ \sqrt{npq}}$, $\sigma =\sqrt{npq}$, then $ \begin{align} M_Z(t)&=e^{-\frac{npt}{\sigma}} (p e^{t/\sigma} + q)^n\\ &=\left[\left(1- \frac{pt}{\sigma}+\frac{p^2t^2}{2\sigma^2}+\ldots\right) \left(1 \mbox{?}+ \frac{pt}{\sigma}+\frac{pt^2}{2σ^2}+\ldots\right)\right]^n\\ &=\left(1+t^2/2n+d(n)/n\right)^n \end{align} $
where $\lim_{n \rightarrow \infty} d(n)=0$, so $\lim_{n \rightarrow \infty} M_Z(t)=e^{\frac{t^2}{2}}$
In here, I can't understand the results of taylor expansion.