# Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 20

The Ratio Test is inconclusive.

#### Work Step by Step

Given $$\sum_{n=2}^{\infty} \frac{1}{\ln n}$$ 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{\ln (n+1)}{\ln n} \\ &= \lim _{n \rightarrow \infty} \frac{\frac{1}{n+1}}{\frac{1}{n}}\\ &= \lim _{n \rightarrow \infty}\frac{n}{n+1}\\ &=1 \end{align*} Thus the Ratio Test is inconclusive.

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