University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.5 - Absolute Convergence; The Ratio and Root Tests - Exercises - Page 515: 9



Work Step by Step

Consider $a_n=\dfrac{7}{(2n+5)^n}$ By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ $l=\lim\limits_{n \to \infty} |a_n|^{1/n}=\lim\limits_{n \to \infty}|\dfrac{\sqrt [n] 7}{{\sqrt [n] {(2n+5)^n}}}|=\lim\limits_{n \to \infty} \dfrac{1}{(2n+5)}$ or, $=\dfrac{1}{\infty}$ so, $l =0\lt 1$ Thus , the given series converges by the Root Test.
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