TITLE: Does the series $\sum\frac{\ln n}{n^{2}}$ Converge?
QUESTION [9 upvotes]: Does the series
$$\sum_{n=2}^{\infty}\frac{\ln n}{n^2}$$
converge?
I'm searching for a solution that does not use the Integral test, Stirling, L’Hôpital or functions theorems.
I tried ratio test, and also comparing it to another series.
Thank you very much.

REPLY [3 votes]: Here is a different style of answer, which may be useful to some.
$\zeta(s)=\sum_{n=1}^\infty n^{-s}$, the Riemann zeta function, is analytic on $\Re(s)>1$.  This is because on any half plane $\Re(s)>1+\epsilon$, it is a uniformely convergent series of analytic functions.  Its derivative, $$\zeta^{'}(s)=-\sum_{n=1}^\infty \log n n^{-s}$$ then also converges on all of $\Re(s)>1$.  Now, notice that your sum is $-\zeta^{'}(2).$