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 547: 32


$$-\frac{4}{15} $$

Work Step by Step

Given $$\sum_{n=0}^{n=\infty} \frac{3(-2)^{n}-5^{n}}{8^{n}}=\sum_{n=0}^{n=\infty} 3 \cdot\left(\frac{-2}{8}\right)^{n}-\sum_{n=0}^{n=\infty}\left(\frac{5}{8}\right)^{n}$$ Since the series is a geometric series with $|r_1|= \frac{2}{8}<1$ and $|r_2|= \frac{5}{8}<1$, then the series converges and has the sum \begin{align*} S&=S_1-S_2\\ &=\frac{a_1}{1-r_1}-\frac{a_1}{1-r_2}\\ &=\frac{3}{1-\frac{-2}{8}}-\frac{1}{1-\frac{5}{8}}\\ &=-\frac{4}{15} \end{align*}
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