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: 31



Work Step by Step

Given $$\sum_{n=0}^{n=\infty} \frac{8+2^{n}}{5^{n}}=\sum_{n=0}^{n=\infty} 8 \cdot\left(\frac{1}{5}\right)^{n}+\sum_{n=0}^{n=\infty}\left(\frac{2}{5}\right)^{n}$$ Since the series is a geometric series with $|r_1|= \frac{1}{5}<1$ and $|r_2|= \frac{2}{5}<1$, 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{8}{1-\frac{1}{5}}+\frac{1}{1-\frac{2}{5}}\\ &= \frac{35}{3} \end{align*}
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