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: 68



Work Step by Step

We are given the geometric series: $\sum_{k=1}^{\infty} 2\left(\dfrac{3}{4}\right)^{k}$ Determine the elements of the geometric series: $a_1=2\left(\dfrac{3}{4}\right)^{1}=\dfrac{3}{2}$ $r=\dfrac{3}{4}$ We compute $|r|$: $|r|=\left|\dfrac{3}{4}\right|=\dfrac{3}{4}$ Because $|r|=\dfrac{3}{4}<1$, the series converges. Determine its sum: $S=\dfrac{a_1}{1-r}=\dfrac{\dfrac{3}{2}}{1-\dfrac{3}{4}}=\dfrac{\dfrac{3}{2}}{\dfrac{1}{4}}=6$
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