Answer
The Ratio Test is inconclusive.
Work Step by Step
Given
$$\sum_{n=2}^{\infty} \frac{1 }{n\ln n}$$
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} \frac{n\ln n }{(n+1)\ln (n+1)} \\
&=\lim _{n \rightarrow \infty} \frac{\ln n }{\ln (n+1)} \lim _{n \rightarrow \infty} \frac{n }{(n+1)} \ \ \cr & \text{Using L'Hopital rule gives:}\\
&=\lim _{n \rightarrow \infty} \frac{\frac{1}{n} }{\frac{1}{n+1}}\\
&=1
\end{align*}
Thus the Ratio Test is inconclusive.
