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 $B_n=\dfrac{\ln n}{n- \ln n}$
a) $\lim\limits_{n \to \infty} p_n=\lim\limits_{n \to \infty}\dfrac{\ln n}{n- \ln n}=\dfrac{\frac{1}{n}}{1-\frac{1}{n}}=0$;
b) Now, $B_n=\dfrac{\ln n}{n- \ln n}$ we can notice from a decreasing sequence that $B'_n=\dfrac{1-\ln n}{(n- \ln n)^2}\lt 0$
Therefore, the given series converges conditionally by the Alternating Series Test.
