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