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Is $\sum_{n=1}^{\infty}\frac{\log n}{n^{2}}$ convergent? How to show that? I was trying to prove Mertens third theorem and i got stuck at this.

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Note that $\log (n) \le n^{0.5}$ and use the comparison test.

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    $\begingroup$ Was writing exactly the same. You win. $\endgroup$ – user88595 Apr 6 '14 at 9:20
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    $\begingroup$ A general claim can be proved: For all $\alpha , \beta , >0$ it holds that $\frac{\log^\alpha (x)}{x^\beta}\xrightarrow{x \to \infty}0$ $\endgroup$ – Amihai Zivan Apr 6 '14 at 9:30
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    $\begingroup$ @AmihaiZivan Yes that has always irked me. I mean $\log$ is asymptotically lower than any positive power of $x$, and as soon as the power reaches $0$, it goes "lol. nope. i win" $\endgroup$ – Guy Apr 6 '14 at 9:42
  • $\begingroup$ Yup, it's definitely a nice result which can be proved quite easily. $\endgroup$ – Amihai Zivan Apr 6 '14 at 9:52

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