Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 10 - Section 10.3 - Geoetric Sequences and Series - Exercise Set - Page 1076: 99

Answer

The sum is $3$. See the graph below:
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Work Step by Step

Consider the given function; $f\left( x \right)=\frac{2\left[ 1-{{\left( \frac{1}{3} \right)}^{x}} \right]}{1-\frac{1}{3}}$ The infinite geometric series is: $2+2\left( \frac{1}{3} \right)+2{{\left( \frac{1}{3} \right)}^{2}}+2{{\left( \frac{1}{3} \right)}^{3}}+\cdots $ From the graph, the horizontal asymptote of the given function is $y=3$. Consider the given series, $2+2\left( \frac{1}{3} \right)+2{{\left( \frac{1}{3} \right)}^{2}}+2{{\left( \frac{1}{3} \right)}^{3}}+\cdots $ The first term of the series is ${{a}_{1}}=2$. And the common ratio is $\begin{align} & r=\frac{1}{3} \\ & <1 \end{align}$ Use the formula for the sum of an infinite geometric series with the first term ${{a}_{1}}$ and common ratio r, $S=\frac{{{a}_{1}}}{1-r}$ Substituting ${{a}_{1}}=2$ and $r=\frac{1}{3}$, the sum of the given infinite geometric series is $\begin{align} & S=\frac{2}{1-\left( \frac{1}{3} \right)} \\ & =\frac{2}{\frac{2}{3}} \\ & =3 \end{align}$ So, the sum of the given infinite series is 3.
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