Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.3 Geometric Sequences; Geometric Series - 12.3 Assess Your Understanding - Page 825: 57


Converges, sum: $\frac{8}{5}$

Work Step by Step

An infinite geometric series converges if and only if $|r|\lt1$, where $r$ is the common ratio. If it converges, then it equals $\frac{a_1}{1-r}$ where $a_1$ is the first term. The common ratio is the quotient of two consecutive terms: $r=\frac{a_2}{a_1}=\dfrac{-\frac{1}{2}}{2}=-\frac{1}{4}$. $|-\frac{1}{4}|=\frac{1}{4}\lt1$, thus it converges. Hence the sum (since $a_1=2$): $\dfrac{2}{1-(-\frac{1}{4})}=\frac{8}{5}$
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