Answer
The series converges for all values of $p$ (can be any real number).
Work Step by Step
Apply the Test of Divergence to find the value of $p$ for the given alternating series.
When $p\leq 0$
Then $\lim\limits_{n \to \infty}\dfrac{(\ln n)^p}{n}=0$ . This means that the limit $0$ satisfies all the conditions for the alternating series test. Thus, the series will converge by the Test of Divergence.
When $p \gt 0$
Then $\lim\limits_{n \to \infty}\dfrac{(\ln n)^p}{n}=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}(1/n)}{1}=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}}{n}$
When $p \leq 1$
Then $\lim\limits_{n \to \infty}\dfrac{(\ln n)^p}{n}=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}(1/n)}{1}=\lim\limits_{n \to \infty}\dfrac{p(\ln n)^{p-1}}{n}=0$
This means that the limit $0$ satisfies all the conditions for the alternating series test. Thus, the series will converge by the Test of Divergence.
Hence, the series converges for all values of $p$ (can be any real number).
