University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 9 - Section 9.1 - Sequences - Exercises - Page 488: 42



Work Step by Step

$a_{n}=\displaystyle \frac{1}{(0.9)^{n}}=\left( \frac{1}{0.9}\right)^{n}$ $=\displaystyle \left( \frac{1}{\frac{9}{10}}\right)^{n}$ $=\displaystyle \left( \frac{10}{9}\right)^{n}$ The function $f(x)=a^{x},\ a\gt 1$ increases without bound when $ x\rightarrow\infty$. Thus, $\displaystyle \lim_{n\rightarrow\infty}\{a_{n}\}$ =$\infty.$ $\{a_{n}\}$ diverges.
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