Answer
The Ratio Test is inconclusive
Work Step by Step
Given
$$\sum_{n=0}^{\infty} \frac{3n+2}{5n^{3}+ 1}$$
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{3n+5}{5(n+1)^{3}+ 1}\frac{5n^{3}+ 1}{3n+2}\\
&=\lim _{n \rightarrow \infty} \frac{\left(3n+5\right)\left(5n^3+1\right)}{\left(5n^3+15n^2+15n+6\right)\left(3n+2\right)}\\
&=1
\end{align*}
Thus the Ratio Test is inconclusive.
