University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 9 - Questions to Guide Your Review - Page 551: 7

Answer

See below.

Work Step by Step

A geometric series is a series of the form $ a+ar+ar^{2}+\displaystyle \cdots+ar^{n-1}+\cdots=\sum_{n=1}^{\infty}ar^{n-1},\quad a\neq 0$. The parameter $r$ is called the common ratio of the series and $a$ is the first term of the sum of the series. It converges to $\displaystyle \frac{a}{1-r}$ if $|r|\lt 1$ and diverges if $|r|\geq 1.$ Example: The series $\displaystyle \sum_{n=1}^{\infty}2\cdot 3^{n-1}$ diverges, because $|r|=|3|\geq 1$ Example $\displaystyle \sum_{n=1}^{\infty}2\cdot(0.5)^{n-1}$ converges because $|r|=|0.5|\lt 1$ and, the sum equals $\displaystyle \frac{2}{1-0.5}=\frac{2}{0.5}=4$
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