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