Answer
Absolutely convergent.
Work Step by Step
The Integral Test states that the series converges when the integral $\int_{k}^\infty f(x) dx$ converges.
Consider the series $a_n=\int_{2}^\infty \dfrac{1}{x(\ln x)^2} dx$
Suppose $p=\ln x$ and $dp=\frac{dx}{x}$
Thus,
$a_n=\int_{\ln 2}^\infty \dfrac{dp}{p^2} dx$
$a_n=-\dfrac{1}{p}|_{\ln 2}^\infty$
$a_n=-\dfrac{1}{\infty}+\dfrac{1}{\ln 2}$
$a_n=\dfrac{1}{\ln 2}$
Therefore, the given series is Absolutely convergent.
