## University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson

# Chapter 9 - Section 9.5 - Absolute Convergence; The Ratio and Root Tests - Exercises - Page 516: 64

#### Answer

Both the root and ratio tests fail.

#### Work Step by Step

Consider $a_n=\dfrac{1}{(\ln n)^p}$ Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{1}{(\ln (n+1))^p}}{\dfrac{1}{(\ln n)^p}}|=\lim\limits_{n \to \infty}|\dfrac{(\ln n)^p}{(\ln (n+1))^p}=1$ Thus, the ratio test does not satisfy. Now, apply the root test. By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{1}{(\ln n)^p})^{1/n}=1$ Thus, the root test also fails.

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