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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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Thank you Brian, taking the sum from $n=1$ to $n=m$ is brilliant! –  Chloe Sep 30 '12 at 19:39
    
@Chloe: You’re welcome! –  Brian M. Scott Sep 30 '12 at 19:43

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