Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Section 10.5 - Absolute Convergence; The Ratio and Root Tests - Exercises 10.5 - Page 597: 13



Work Step by Step

Let us consider $a_n=\dfrac{8}{(3+\dfrac{1}{n})^{2n}}$ In order to solve the given series we will take the help of Root Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges. In order to solve the series we will take the help of Root Test. It can be defined as follows: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{8}{(3+\dfrac{1}{n})^{2n}}|}$ $\implies \lim\limits_{n \to \infty} \dfrac{8}{(3+\dfrac{1}{n})^{2}}=\dfrac{8}{(3)(3)}=\dfrac{8}{9} \lt 1$ Hence, the series Converges by the Root Test.
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