Answer
Divergent
Work Step by Step
Let us consider $f(x)=\dfrac{\ln x}{x}$
Here, the function $f(x)$ is positive, continuous for $x \geq 2$ and $f(x)$ is decreasing for $x \gt 3$
Then $\int_3^\infty \dfrac{\ln x}{x}dx= \lim\limits_{k \to \infty} \int_3^k \dfrac{\ln x}{x}dx=\lim\limits_{k \to \infty} [(\dfrac{1}{2})(\ln^2 x)]_3^k$
and $ \lim\limits_{k \to \infty} [(\dfrac{1}{2})(\ln^2 (k)-\ln^2 (3))]= \infty$
Thus, the sequence $\Sigma_{n=2}^\infty \dfrac{\ln n}{n}$ is Divergent.
