Answer
$P \gt 0$
Work Step by Step
Apply the Test of Divergence to find the value of $p$ for the given alternating series.
1. when $p\lt 0$
Then $\lim\limits_{n \to \infty}(-1)^{n-1} n^{|p|}=$ Limit does not exist. Thus, the series will not converge by the Test of Divergence.
2. when $p = 0$
Then $\lim\limits_{n \to \infty}(-1)^{n-1} n^{0}=$ Limit does not exist. Thus, the series will not converge by the Test of Divergence.
3. when $p\gt 0$
Then $\lim\limits_{n \to \infty}\dfrac{1}{n^p}=0$ . 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.
