Prove that $\sum_{n=1}^\infty \frac{x^{2n}}{(n+x)^2}$ converges uniformly on $S=[0,1]$

Question

Can someone verify my proof this is what I have so far?

Proof using the Weirstrass M. test. Assume $x \in S,$ that is, $[0,1].$ Then $$|f_n(x)|= \frac{x^{2n}}{(n+x)^2} \leq \frac{1}{n^2}$$ $$\leq \frac{1}{n^2} = M_n$$ for all $x \in S$ and all $n \in \mathbb{N}.$ Hence $$\sum^\infty _{n=1} M_n= \sum \frac{1}{n^2}$$ converges by the $p-\text{series}$ test.