Show that $\sum_{n=3}^{\infty}f_n$ is uniformly convergent in $(-\pi,\pi)$ for $f_n := \ln(\cos(\frac{x}{n}))$
Simply use $$\log(x) \le x-1$$ for $x>0$. So you get $$|\log(\cos(x/n))| \le 1- \cos(x/n).$$ Now use $\cos(x)=1+O(x^2)$ to get $$|\log(\cos(x/n))| \le O(x^2/n^2).$$ Hence, $$\sum_{n=3}^\infty |\log(\cos(x/n))| < \infty$$ for $x \in [-\pi,\pi]$. Since $|\log(\cos(x/n))|$ is maximal for $x=\pi$ this implies also uniform convergence.