Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{\ln n}{n}$
Here, we have $f(n)=\dfrac{\ln n}{n}$
and $f'(n)=\dfrac{(1-\ln n)}{n^2} \leq 0$ when $n \geq 3$
The negative sign implies that the sequence $u_n$ is decreasing.
Thus, $\lim\limits_{n \to \infty} u_n=\lim\limits_{n \to \infty}\dfrac{\ln n}{n}=\dfrac{\infty}{\infty}$
We will have to apply L-Hospital's rule.
$\implies \lim\limits_{n \to \infty}\dfrac{\dfrac{1}{n}}{1}=0$
Hence, the series converges by the Alternating Series Test.