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 - Preliminary Questions - Page 568: 2


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Work Step by Step

We have to find $\rho$ $$\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty}\left|\frac{1/2^{n+1}}{1/2^n}\right|=\frac{1}{2}\lt1$$ Hence the Ratio Test is conclusive for $\frac{1}{2^n}$. $$\rho= \lim _{n \rightarrow \infty}\left|\frac{1/(n+1)}{1/n}\right|= \lim _{n \rightarrow \infty}\frac{n}{n+1}=1$$ Hence the Ratio Test is inconclusive for $\frac{1}{n}$.
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