In this question, the solver depended on a way I didn't use to solve in.
I thought of finding a bound for $|R_n(x)|$ that's going to $0$ in order to prove the uniform convergence. But I always find a problem dealing with $(-1)^n$. I think of using the fact that in alternating convergent series, the first term dominates.
$|R_n(x)| = |\sum_{k=n+1}^\infty (-1)^k\frac{x^2 + k}{k^2}| \leq |(-1)^{n+1}\frac{x^2+n+1}{(n+1)^2}|= \frac{x^2+n+1}{(n+1)^2}$ but then,
$|R_n(n)| \leq \frac{n^2+n+1}{(n+1)^2} \longrightarrow 1$ not $0$. So for sure I'm doing mistakes in estimating, since the answer is "uniformly convergent".
Guys, I'm just a beginner, you can call me a self learner. Please help me as I feel I won't be good at such problems :(