Answer
The Ratio Test is inconclusive for the $p-$series.
Work Step by Step
Given
$$\sum_{n=1}^{\infty}\frac{1}{n^p}$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{n^p}{(n+1)^p} \right|\\
&= \lim _{n \rightarrow \infty} \left(\frac{n}{n+1}\right)^p \\
&=\left( \lim _{n \rightarrow \infty} \frac{n}{n+1}\right)^p \\
&= 1
\end{align*}
Thus, Ratio Test is inconclusive for the $p-$series.
