Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 32


The series $\Sigma_{n=1}^{\infty}4^na_n$ diverges.

Work Step by Step

Let $b_n=4^na_n$; then applying the ratio test, we have $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{b_{n+1}}{b_{n}}\right|=\lim _{n \rightarrow \infty} \frac{4^{n+1}}{4^n}\left|\frac{a_{n+1}}{a_{n}}\right|=\frac{4}{3}\gt1 $$ Hence, the series $\Sigma_{n=1}^{\infty}4^na_n$ diverges.
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