Answer
Converges conditionally
Work Step by Step
Alternating Series Test states that consider a series $\Sigma a_n$ such that $p_n=(-1)^n q_n$; $q_n \geq 0$ for all $n$
When the following conditions are satisfied then the series converges such that
i) $\lim\limits_{n \to \infty} q_n=0$;
ii) $q_n$ is a decreasing sequence.
Here, we have $q_n=\dfrac{1}{n \ln n}$
i) Now, $\lim\limits_{n \to \infty} q_n=\lim\limits_{n \to \infty}\dfrac{1}{(n)(\ln n)}=\dfrac{1}{\infty}=0$;
ii) Also, $q_n=\dfrac{1}{n \ln n}$ shows a decreasing sequence as the function $n \ln n$ increases.
Hence, the given series Converges conditionally by the Alternating Series Test.
