Chapter 12.4, Problem 69E

To determine

Whether the sequence $a_n=\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}$ converges or diverges by creating a scatter plot of the sequence.

Answer to Problem 69E

The given sequence $a_n=\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}$ diverges.

Explanation of Solution

Given information :

Given sequence $a_n=\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}$

To find the limit of $a_n=\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}$ by algebraically:

  $\begin{array}{l}\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}\\ \text{since}1.5>1,So{\left(1.5\right)}^n\to \infty \text{as}n\to \infty .\\ \therefore \underset{n\to \infty }{\lim }{a}_n=3\times \infty =\infty .\end{array}$

Graph: The scatter plot of $a_n=\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}$ is

  

Interpretation: from above graph, we observe that the sequence $a_n$ increase without bound as $n$ increases towards positive direction of $X-axis$ .

Thus, the sequence is not approaching to a unique finite value as $n\to \infty .$

  $\therefore \underset{n\to \infty }{\lim }{a}_n=\infty$

Hence, the given sequence $a_n=\frac{3\left[1-{\left(1.5\right)}^n\right]}{1-1.5}$ diverges.