# Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 4

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.

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