# Computing $S_n(x)$, the partial sum of a series explicitly

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?

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