Central Limit Theorem Characteristic function convergence

Question

So I was recently looking through a proof of the central limit theorem using the expansion of characteristic function, and came to a point where the step was to show that the $$\lim_{n\to\infty}(1-\frac{t^2}{2n}+O(\frac{1}{n^{3/2}}))^n=\exp[{{\frac{-t^2}{2}}]}$$ This expansion is the representation of $E[\exp(\frac{itx}{\sqrt{n}})]$ with the first two moments plugged in $(0,1)$. I see how the first two elements of the limit argument converge exactly to the answer, but am struggling to come to a proof of how the entire argument does. So far I've come to the idea of using the binomial expansion by grouping the first two terms separate from the rest of the infinite series e.g. $$\lim_{n\to\infty}[(1-\frac{t^2}{2n})^n+n(1-\frac{t^2}{2n})^{n-1}(f(n))+\cdots]$$ where $f(n)$ is standing in for the infinite series involving larger and larger denominator values of $n$, but am completely lost. I am looking to see if someone could tell me if I'm headed in the right direction, or if I'm completely off the mark.

Answer 1

The justification of the result $$ \lim_{n\to\infty}(1-\frac{t^2}{2n}+O(\frac{1}{n^{3/2}}))^n=\exp[{{\frac{-t^2}{2}}]} $$ is more subtle than it looks because the characteristic function involves complex numbers. So this isn't a real analysis problem! You need a theorem about a sequence of complex numbers converging to an exponential. This one should work: If $z_n \to z$ then $(1+z_n/n)^n \to e^z$ .