Answer
Divergent
Work Step by Step
Consider $f(x)=\dfrac{\ln x}{x}$
Now, take the integral test to find the convergence and divergence for the sequence.
We have $\int_2^\infty \dfrac{\ln x}{x}dx= \lim\limits_{a \to \infty} \int_2^a \dfrac{\ln x}{x}dx$
or, $ \lim\limits_{a \to \infty} [\dfrac{1}{2}(\ln^2 x)]_2^a=\lim\limits_{a \to \infty} [\dfrac{1}{2}(\ln^2 a-\ln^2 2)]= \infty$
Hence, the sequence $\Sigma_{n=2}^\infty \dfrac{\ln n}{n}$ is Divergent.
