Answer
Ratio Test is inconclusive.
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{\sqrt{n+1}}{n^{8}}$$
By using Ratio the test
\begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}}{(n+1)^{8}} \frac{n^{8}}{\sqrt{n+1}}\\ &=1 \end{align*}
Thus, the Ratio Test is inconclusive.
