Chapter 9.3, Problem 82E

To determine

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

Answer to Problem 82E

The sum of the series is $S_n={\displaystyle \sum }_{n=0}^{2}2{\left(\frac{4}{5}\right)}^n$

Explanation of Solution

Given:

  $f(x)=2\left[\frac{1-{(0.8)}^x}{1-(0.8)}\right]$

Consider the function,

  $f(x)=2\left[\frac{1-{(0.8)}^x}{1-(0.8)}\right]$

Enter the equation in T1-83

  

Click on WINDOW button and set the minimum and maximum values of labels (axis)

  

Click on the GRAPH button then below screen will be obtained,

  

Hence the Horizontal asymptote is $y=10$ .

The series is in the form of sum of geometric finite series that is,

  $S_m=a\left[\frac{1-r^{\ast }}{1-r}\right]$

Implies that $a=2,r=0.8$

Then the $n^{\text{th}}$ term is,

  $\begin{array}{cc}{a}_x& =a{r}^x\\ & =2{(0.8)}^x\\ & =2{\left(\frac{8}{10}\right)}^x\\ & =2{\left(\frac{4}{5}\right)}^a\end{array}$

Hence the sum of the series is $S_n={\displaystyle \sum }_{n=0}^{2}2{\left(\frac{4}{5}\right)}^n$