Let $Z_k$ be a sequence of independent random variables so that $P(Z_k=1)=1/2,P(Z_k=-1)=1/2$ for every $k \in \mathbb{N}$. Also let $S_n=\sum_{k=1}^n Z_k,G_n=\frac{S_n}{\sqrt{n}}$. Show that the characteristic function of $G_n$ is
$$\psi_{G_n}(u)=\cos(\frac{u}{\sqrt{n}})^n$$. I know that $\psi_X(u):=E(e^{iux})$ and I know that the expected value can be written as an integral but I'm stuck on doing this.
Exploit the following rules:
If $X$ and $Y$ are independent rv's then:$$\psi_{X+Y}(t)=\mathbb Ee^{it(X+Y)}=\mathbb Ee^{itX}e^{itY}=\mathbb Ee^{itX}\mathbb Ee^{itY}=\psi_X(t)\psi_Y(t)\tag1$$(this to find the characteristic function of $S_n$)
And for a constant $a$:$$\psi_{aX}(t)=\mathbb Ee^{itaX}=\psi_X(at)\tag2$$(this to find the characteristic function of $G_n$)