Given $X_1, X_2, ..., X_n$ $i.i.d$ with mean $\mu$ , variance $\sigma^2$ and MGF $X_j = M$ and $Z_n = \sqrt{n}\big(\frac{\tilde X_n - \mu}{\sigma}\big)$
Find MGF of $Z_n$ in terms of $M$.
What I managed to do is the following:
$$MGF_{Z_n}(t) = E\big(e^{t\sqrt{n}\big(\frac{\tilde X_n - \mu}{\sigma}\big)}\big) = $$ $$ E\big(e^{\frac{t\sqrt n}{\sigma}(\tilde X - \mu)}\big) = e^{-\mu \frac{t\sqrt n}{\sigma}}E\big(e^{\frac{t\sqrt n}{\sigma}\tilde X}\big ) =$$ $$ e^{-\mu \frac{t\sqrt n}{\sigma}}E\big(e^{\frac{t}{\sigma\sqrt n}(X_1 + ... + X_N)}\big) = $$ $$e^{-\mu \frac{t\sqrt n}{\sigma}}E\big(e^{\frac{t}{\sigma\sqrt n}X_i}\big)^n $$
I don't know how to go from $E\big(e^{\frac{t}{\sigma\sqrt n}X_i}\big) $ to M which is $E(e^{tX_i})$. I'd appreciate some help.