Answer
$p\gt 0$
Work Step by Step
Case 1: Use the Test of Divergence to calculate the value of $p$ for the given alternating series when $p\lt 0$
Now, $\lim\limits_{n \to \infty}(-1)^{n-1} n^{|p|}=DNE$
So, the series will not converge by the Test of Divergence.
Case 2: Use the Test of Divergence to calculate the value of $p$ for the given alternating series when $p = 0$
Now, $\lim\limits_{n \to \infty}(-1)^{n-1} n^{0}=DNE$ . So, the series will not converge by the Test of Divergence.
Case 3: Use the Test of Divergence to calculate the value of $p$ for the given alternating series when $p\gt 0$
Now, $\lim\limits_{n \to \infty}\dfrac{1}{n^p}=0$ .
Here, the limit $0$ satisfies all the conditions for the alternating series test and so, the given series will converge for the value of $p\gt 0$ by the Test of Divergence.