Chapter 9.3, Problem 81E

To determine

Sketch the graph and identify the asymptote function of the graph and determine its relationship to the sum.

Answer to Problem 81E

The sum is equal to the asymptote of the given function

Explanation of Solution

Given:

  $f(x)=6\left[\frac{1-{(0.5)}^x}{1-0.5}\right]$

The graph of the function $f(x)=6\left[\frac{1-{(0.5)}^x}{1-0.5}\right]$ is,

  

From the figure, it is seen that the horizontal asymptote of the function is,

  $y=12$

The infinite sum of the sequence ${\displaystyle \sum }_{n=0}^{\infty }6{\left(\frac{1}{2}\right)}^n$ is,

The common ratio of this geometric series is,

  $\frac{1}{2}$ The first term is $6.$

Therefore, the sum is,

  $\begin{array}{cc}{S}_x& =\frac{a_1}{1-r}\\ & =\frac{6}{1-\frac{1}{2}}\\ & =12\end{array}$

The sum is equal to theasymptote of the given function.