Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.2 Summing and Infinite Series - Exercises - Page 548: 35



Work Step by Step

Given $$\frac{7}{8}-\frac{49}{64}+\frac{343}{512}-\frac{2401}{4096}+\cdots=\sum_{n=0}^{n=\infty} \frac{7}{8}\left(\frac{-7}{8}\right)^{n} $$ Since the series is a geometric series with $|r|= \frac{7}{8}<1$, then the series converges and has the sum \begin{align*} S &=\frac{a_1}{1-r} \\ &=\frac{\frac{7}{8}}{1-\frac{-7}{8}} \\ &=\frac{7}{15} \end{align*}
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