Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - 11.2 Exercises - Page 735: 17

Answer

Divergent

Work Step by Step

We are given the series: $3-4+\dfrac{16}{3}-\dfrac{64}{9}+....$ We determine the ratios of consecutive terms: $\dfrac{-4}{3}=-\dfrac{4}{3}$ $\dfrac{\dfrac{16}{3}}{-4}=-\dfrac{16}{3}\cdot\dfrac{1}{4}=-\dfrac{4}{3}$ $\dfrac{-\dfrac{64}{9}}{\dfrac{16}{3}}=-\dfrac{64}{9}\cdot\dfrac{3}{16}=-\dfrac{4}{3}$ As all ratios are the same, the series is a geometric series. Its ratio is $r=-\dfrac{4}{3}$. Because $|r|=\left|-\dfrac{4}{3}\right|=\dfrac{4}{3}>1$, the series is divergent.
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