Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 14 - Sequences, Series, and the Binomial Theorem - 14.3 Geometric Sequences and Series - 14.3 Exercise Set - Page 912: 43

Answer

Yes, the infinite geometric series has a limit and the value of the limit is ${{S}_{\infty }}=27$.

Work Step by Step

$18+6+2+\cdots $ Here, ${{a}_{1}}=18$,${{a}_{2}}=6$ The value of $\left| r \right|$ is, $\begin{align} & \left| r \right|=\left| \frac{6}{18} \right| \\ & =\frac{1}{3} \end{align}$ Thus, the series does have a limit. Find the limit of the infinite geometry series for the formula ${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$. $\begin{align} & {{S}_{\infty }}=\frac{18}{1-\frac{1}{3}} \\ & =\frac{18}{\frac{2}{3}} \\ & =\frac{3\cdot 18}{2} \\ & =27 \end{align}$ Therefore, the limit of the infinite geometric series is ${{S}_{\infty }}=27$.
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