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

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.