Answer
Divergent
Work Step by Step
Let us consider $f(x)=\dfrac{\ln x}{\sqrt x}$
Here, the function $f(x)$ is positive, continuous for $x \geq 2$ and $f(x)$ is decreasing for $x \gt 3$.
Then $\lim\limits_{k \to \infty} \int_2^k \dfrac{\ln x}{\sqrt x}dx= \lim\limits_{k \to \infty} [(2) (\ln x)e^{\ln x/2} -(4) (e^{\ln x/2})]_{\ln 2}^k$
and $\lim\limits_{k \to \infty} [(2) (\ln x) (e^{\ln x/2}) -(4) (e^{\ln x/2})]_{\ln 2}^k=\infty$
Thus, the sequence $\Sigma_{n=2}^\infty \dfrac{\ln n}{\sqrt n}$ is Divergent.
