Given the sum $$ S = \sum_{n=1}^\infty \frac{1}{n} \frac{1}{(2+x^2)^n} $$ I have to show that $S$ converges uniformly for all $x \in \mathbb{R}$. I know that I probably have to use Weisterstrass' M-test to find a sequence of positive numbers ${M_n}$ satisfying that $ |f_n(x)| \leq M_n$ and that $\sum_{n=1}^\infty M_n < \infty$ but I am not sure whether my solution is correct. Do you mind verifying?
As $x \in \mathbb{R}$ we must have that $$ \frac{1}{(2+x^2)^n} \leq \frac{1}{n} $$ for all $n \in \mathbb{N}$. This means that $$ \frac{1}{n} \frac{1}{(2+x^2)^n} \leq \frac{1}{n^2} $$ and from analysis we know that $$ \sum_{n=1}^\infty \frac{1}{n^2} $$ converges. Thus Weiterstrass' M-test gives that S converges uniformly. Is this approach ok?