Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 11 - Section 11.6 - Absolute Convergence and the Ratio and Root Tests - 11.6 Exercises - Page 778: 28

Answer

Absolutely comvergent

Work Step by Step

$a_n=(\frac{1-n}{2+3n})^n$ $|a_{n}| = \biggl|(\frac{1-n}{2+3n})^n\biggr|$ By Root Test, we get $\lim_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}\biggl|\frac{1-n}{2+3n}\biggr|=\biggl|-\frac{1}{3}\biggr|=\frac{1}{3}$ Since the limit $ < 1 $, the series is absolutely convergent
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