I'm trying to show that the integral $ \int_1^\infty \frac{ (\log(y))^n }{y^2} \ dy $ is convergent for every real number $ n \geq 1$. If $ n < 2$, I can bound $ |\log(y)|$ by $y$ and hence show that the integral does converge, but I'm not sure how to construct a tighter bound for the case $ n \geq 2$.
Any help is appreciated!