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: 58



Work Step by Step

We have $S_n=a_1+a_2+......+(-1)^{n+1}a_n$ Now, $|S-S_n| \leq |a_{n+1}| \implies \space |Error| \lt |a_{n+1} |$ or, $|\dfrac{1}{k!}| \lt \dfrac{5}{10^6}$ or, $k \ge 9$ Thus, we need at least 9 terms. So, $S \approx S_8 =1-1+\dfrac{1}{2!}-\dfrac{1}{3!}+.... \approx 0.3678819444$
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