Answer
Converges conditionally
Work Step by Step
The Alternating Series Test states:
Consider a series $\Sigma a_n$ such that $A_n=(-1)^n B_n$; $B_n \geq 0$ for all $n$
If the following conditions are satisfied, then the series converges:
a) $\lim\limits_{n \to \infty} B_n=0$;
b) $B_n$ is a decreasing sequence.
We notice that $B_n=\dfrac{1}{n \ln n}$
a) Now, $\lim\limits_{n \to \infty} q_n=\lim\limits_{n \to \infty}\dfrac{1}{(n)(\ln n)}=\dfrac{1}{\infty}=0$;
b) Also, $q_n=\dfrac{1}{n \ln n}$ shows a decreasing sequence as the function $n \ln n$ increases.
Therefore, the given series converges conditionally by the Alternating Series Test.