Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{4}{ (\ln n)^2}$
Here, $a_n$ is positive for all the values of $n$.
Now, $f(n)=\dfrac{4}{ (\ln n)^2}$
and $f'(n)=\dfrac{-8}{n(\ln n)^4} \lt 0$
The negative sign shows that, the sequence $u_n$ is not increasing.
Thus, $\lim\limits_{n \to \infty} u_n=\dfrac{4}{ (\ln n)^2}=0$
Hence, by the Alternating Series Test, the series converges.