Answer
The series converges for all the values of $p$ (can be any real number).
Work Step by Step
Case 1: When $p\leq 0$
$\lim\limits_{n \to \infty}\dfrac{(\ln n)^p}{n}=0$ .
Thus, the series converges by the Test of Divergence.
Case 2: When $p \gt 0$
$\lim\limits_{n \to \infty}\dfrac{(\ln n)^p}{n}=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}(1/n)}{1}$
or, $=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}}{n}$
Case 3: When $p \leq 1$
$\lim\limits_{n \to \infty}\dfrac{(\ln n)^p}{n}=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}(\dfrac{1}{n})}{1}$
and $\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}}{n}=0$
This implies that the limit $0$ satisfies all the conditions for the alternating series test and so, the series converges by the Test of Divergence.
Hence, the series will converge for all values of $p$ (any real number).
