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: 35


The Ratio Test is inconclusive for the $p-$series.

Work Step by Step

Given $$\sum_{n=1}^{\infty}\frac{1}{n^p}$$ By using the Ratio Test, we get: \begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|\frac{n^p}{(n+1)^p} \right|\\ &= \lim _{n \rightarrow \infty} \left(\frac{n}{n+1}\right)^p \\ &=\left( \lim _{n \rightarrow \infty} \frac{n}{n+1}\right)^p \\ &= 1 \end{align*} Thus, Ratio Test is inconclusive for the $p-$series.
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