College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 8, Sequences and Series - Section 8.3 - Geometric Sequences - 8.3 Exercises - Page 615: 68


convergent; $S_{\infty}=\dfrac{2}{3}$

Work Step by Step

RECALL: (1) The sum of an infinite geometric series is convergent if $|r| < 1$. The sum is given by the formula: $S_{\infty}=\dfrac{a}{1-r}$ (2) An infinite geometric series is divergent if $|r|>1$. (3) The common ratio $r$ of a geometric series can be found by dividing any term by the term before it: $r = \dfrac{a_n}{a_{n-1}}$ Solve for $r$ to obtain: $\require{cancel} r = \dfrac{\frac{4}{25}}{\frac{2}{5}} \\r=\dfrac{4}{25} \cdot \dfrac{5}{2} \\r=\dfrac{\cancel{4}2}{\cancel{25}5} \cdot \dfrac{\cancel{5}}{\cancel{2}} \\r=\dfrac{2}{5}$ Since $|\frac{2}{5}|<1$, then the series converges. Solve for the sum using the formula above, with $a=\dfrac{2}{5}$ and $r=\frac{2}{5}$, to obtain: $\require{cancel} S_{\infty}=\dfrac{\frac{2}{5}}{1-\frac{2}{5}} \\S_{\infty}=\dfrac{\frac{2}{5}}{\frac{5}{5}-\frac{2}{5}} \\S_{\infty}=\dfrac{\frac{2}{5}}{\frac{3}{5}} \\S_{\infty} = \dfrac{2}{5} \cdot \frac{5}{3} \\S_{\infty} = \dfrac{2}{\cancel{5}} \cdot \dfrac{\cancel{5}}{3} \\S_{\infty}=\dfrac{2}{3}$
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