Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - Review - True-False Quiz - Page 824: 4



Work Step by Step

If $ \sum c_{n}6^{n}$ is convergent then $\lim\limits_{n \to \infty}c_{n}6^{n}=0$ For limit to be zero, $c_{n}$ must be less than $\frac{1}{6^{n}}$ For $ \sum c_{n}(-2)^{n}$, since it is an alternating series then $\lim\limits_{n \to \infty}c_{n}(-2)^{n}=0$.Since, we know that $c_{n}\lt\frac{1}{6^{n}}$ thus, $c_{n}\lt\frac{1}{(-2)^{n}}$ and limit does go to zero . Therefore, $ \sum c_{n}(-2)^{n}$ is convergent by the Alternating Series Test. Hence, the statement is true.
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