Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.3 Geometric Sequences; Geometric Series - 11.3 Assess Your Understanding - Page 845: 62


Series Converges. and $S_{\infty}=12$

Work Step by Step

The common ratio of a geometric sequence is equal to the quotient of any term and the term before it: $ \ r = \dfrac{a_n}{a_{n-1}}$ or, $r=\dfrac{a_2}{a_1}$ The sum of a convergent infinite geometric series is given by the formula: $S_{\infty}=\dfrac{a_1}{1-r}$ and a geometric series converges if $|r| \lt 1$. where $r \ =common \ ratio \ =\dfrac{1}{3}$ Since $r=|\dfrac{1}{3}| \lt 1$, so the infinite geometric Series Converges. First term: $a_1=8(\dfrac{1}{3})^{1-1}=(8)(1)=8$ Next, $S_{\infty}=\dfrac{8}{1-\dfrac{1}{3}}=\dfrac{8}{2/3}$ Therefore, the sum of the convergent infinite geometric series is: $S_{\infty}=12$
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