Compute explicitly $S_n(x)$, the $n^{th}$ partial sum of the series
$$\sum_{k=1}^∞ \frac{x\left[-1+4k(k+1)x^2\right]}{(1+4k^2x^2)(1+4(k+1)^2x^2)}$$
then compute the sum $S(x)$ of the infnite series, and prove that, for $a > 0$, the series is not uniformly convergent on $(a, a)$, but is uniformly convergent on $(a, ∞)$
My attempt:
$$S_n(x) = \sum_{k=1}^∞ \frac?{1+4k^2x^2} - \frac?{1+4(k+1)^2x^2}$$ which is a telescoping series.
And then, having formed $S_n(x)$, I find $S(x)$ = $\lim_{n→∞} S_n(x)$.
Finally, I find $M_n = \sup|S_n(x) - S(x)|$ and if $\lim_{n→∞} M_n$ = $0$, then it converges uniformly. My problem is in the first step. I don't know how to compute $S_n(x)$ explicitly. Any help please?