Answer
$p\gt 0$
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 $p\lt 0$
Then $\lim\limits_{n \to \infty}(-1)^{n-1} n^{|p|}=$ Limit does not exist.This means that the series will not 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 $p = 0$
Then $\lim\limits_{n \to \infty}(-1)^{n-1} n^{0}=$ Limit does not exist.This means that the series will not converge by the Test of Divergence.
3. We will have to apply the Test of Divergence to find the value of $p$ for the given alternating series when $p\gt 0$
Here, $\lim\limits_{n \to \infty}\dfrac{1}{n^p}=0$
This means that the limit $0$ satisfies all the condition for alternating series test.This means that the series will converge by the Test of Divergence.
