sequences and series - Prove that $sumlimits_{n=1}^infty(-1)^nfrac{x^2+n}{n^2}$ converges uniformly, but not absolutely

My Work:
i) Uniform Convergence (By Weierstrass M-Test):
I am attempting to show that the series converges uniformly on the interval $I=[-a,a]$, in which:$$\sum_{n=1}^\infty(-1)^n\frac{x^2+n}{n^2}\le\sum_{n=1}^\infty(-1)^n\frac{a^2+n}{n^2}=\varepsilon_0$$
Suppose that $\sum\limits_{n=1}^\infty\frac{x^2+n}{n^2}$ will be called "A" and $\varepsilon_0$ will be called "B". By the M-test, if B converges, then A converges uniformly on the interval $I$ defined above.
I am having difficulty proving that B converges, however. I have tried both the root and ratio tests, and they have been unhelpful. For this segment, could someone confirm my logic / provide a hint towards the convergence of B?
ii) Absolute Convergence:
I believe that this function does not converge absolutely because$\sum\limits_{n=1}^\infty\frac{x^2+n}{n^2}$ is a divergent sum as n approaches infinity. Bit of a trivial solution here but I believe it's sufficient.