Teacher says this converges but I keep getting divergence. What am I missing or messing up on?

$\displaystyle \sum_{n=2}^\infty\frac{ln~n}{n^3}$

This is what I get by using integration by parts:

$\displaystyle \int_2^\infty\frac{ln~x}{x^3}~dx=\frac{-ln~x}{2x^2}+\frac{1}{2}\int_2^\infty\frac{dx}{x^3} =\frac{-ln~x}{2x^2}-\frac{1}{4x^2}\big|_2^\infty$

I come up with negative infinity plus a constant = infinity