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?

  • $\begingroup$ Well what do you get for the numerators? And a partial sum has a finite upper bound. $\endgroup$ – Simply Beautiful Art May 10 at 12:23
  • $\begingroup$ @SimplyBeautifulArt I don't know, that's why I put question marks $\endgroup$ – pascale bou chahine May 10 at 12:52

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