University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.6 - Alternating Series and Conditional Convergence - Exercises - Page 521: 50



Work Step by Step

The Alternating Series Estimation Theorem states: Consider a series $\Sigma a_n$ such that $A_n=(-1)^n B_n$; $B_n \geq 0$ for all $n$ If the following conditions are satisfied, then the series converges: a) $\lim\limits_{n \to \infty} B_n=0$; b) $B_n$ is a decreasing sequence. We have $S_n=a_1-a_2+......+(-1)^{n+1}a_n$ Now, $|S-S_n|=\leq |a_{n+1}|$ So, $|\space Error |=|S-S_4| \leq |a_{4+1}|=|a_5|$ or, $=|(-1)^5 \times \dfrac{1}{10^5}|$ or, $=10^{-5}$
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