Answer
Ratio Test is inconclusive.
Work Step by Step
Given
$$\sum_{n=4}^{\infty} \frac{\ln n}{n^{3 / 2}}$$
By using the Ratio test
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{\ln (n+1)}{(n+1)^{3 / 2}} \frac{n^{3 / 2}}{\ln n}\\
&=\lim _{n \rightarrow \infty} \frac{\ln (n+1)}{\ln n} \lim _{n \rightarrow \infty} \frac{n^{3 / 2}}{(n+1)^{3 / 2}} \\
&= \lim _{n \rightarrow \infty} \frac{\frac{1}{n+1}}{ \frac{1}{n}}\\
&=1
\end{align*}
Thus the Ratio Test is inconclusive.
