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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.

Semsem
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esege
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1 Answers1

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

user10444
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    Was writing exactly the same. You win. – user88595 Apr 06 '14 at 09:20
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    A general claim can be proved: For all $\alpha , \beta , >0$ it holds that $\frac{\log^\alpha (x)}{x^\beta}\xrightarrow{x \to \infty}0$ – Amihai Zivan Apr 06 '14 at 09:30
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    @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" – Guy Apr 06 '14 at 09:42
  • Yup, it's definitely a nice result which can be proved quite easily. – Amihai Zivan Apr 06 '14 at 09:52