Let $X_n$ be an iid sequence of random variable with uniform distribution on $(-1,1)$. Using characteristic functions prove that $X_1+X_2+...+X_n$ has density
$$f(x)= \frac{1}{\pi} \int_{0}^{\infty}{(\frac{\sin(t)}{t})^n \cos(tx)dt}$$
I proved that the characteristic function of $X_n$ is $\phi(t)= (\frac{\sin(t)}{t}) $ Thus the characteristic function of $S_n$ is $(\frac{\sin(t)}{t})^n $
I don't know how this could work. Please help me!
Hint: the characteristic function of $X_1+\dots+X_n$ is indeed $\left(\frac{\sin t}t\right)^n$. The inversion formula for an integrable characteristic function ($n\geqslant 2$) will give the result (cut the integral $\int_{-\infty}^0+\int_0^\infty$ and use the substitution $s=-t$ in the first integral).