# Is $\sum_{n=1}^{\infty}\frac{\log n}{n^{2}}$ convergent? How to show that?

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.

Note that $\log (n) \le n^{0.5}$ and use the comparison test.
• 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 6 '14 at 9:30
• @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 6 '14 at 9:42