Answer
The Ratio Test is inconclusive.
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n}{n^{2}+ 1}$$
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+1}{(n+1)^{2}+ 1} \frac{n^{2}+ 1}{n}\\
&=\lim _{n \rightarrow \infty} \frac{\left(n+1\right)\left(n^2+1\right)}{n\left(n^2+2n+2\right)}\\
&= \lim _{n \rightarrow \infty} \frac{n^3+n+n^2+1}{\left(n^3+2n^2+2n\right)}\\
&=1
\end{align*}
Thus the Ratio Test is inconclusive.
