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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!

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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.

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    @Chloe: You’re welcome!2012-09-30