IMC 1996 Problem 11

i) Prove that $$\lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}$$ . ii) Prove that there is a positive constant $c$ such that for every $x\in [1,\infty)$ we have $$\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}$$