Uniform convergence of $ f_n(x)$ = $ \left(\frac{1-\cos x}{x^{\alpha}}\right)^n $ for $ x \in (0,+\infty) $ and $\alpha >0$

Question

Uniform convergence of $ f_n(x)$ = $ \left(\frac{1-\cos x}{x^{\alpha}}\right)^n $ for $ x \in (0,+\infty) $ and $\alpha >0$.

The problem is to find $\alpha$ so that $f_n(x)$ converges uniformly in $(0,+\infty)$. I started looking for the limits of $f_n(x)$ for $x \rightarrow 0$ and for $x \rightarrow +\infty$: I found out that must be $0<\alpha<2$, but I can't proceed with the proof that for those $\alpha$ $f_n(x)$ converges uniformly in $(0,+\infty)$. Can anyone show me how?

Thanks a lot.