Limit of a series

Question

Could anyone help me showing that $$\lim_{x\to\infty}\sum_{n=1}^{\infty}\frac{x}{n(x+n)}$$ does not exist? I know that the sum is converge for each $x$, $$\sum_{n=1}^{\infty}\frac{x}{n(x+n)}\leq \sum_{n=1}^{\infty}\frac{x}{n^{2}}= x\frac{\pi^{2}}{6}$$ but this doesn't help in showing that the above limit doesn't exist!

Answer 1

HINT: If $x>m$, then $$\sum_{n=1}^\infty\frac{x}{n(x+n)}>\sum_{n=1}^m\frac{m}{2mn}=\frac12H_m\;,$$

where $H_m=\sum_{k=1}^m\frac1k$ is the $m$-th harmonic number.