Chapter 11.4, Problem 70E

To determine

To find: Whether the sequence converges or diverges and if the sequence converges find the limit.

Answer to Problem 70E

The given sequence converges to $6$ .

Explanation of Solution

Given Data:

The given sequence is,

  ${\mathrm{a}}_{\mathrm{n}}=\frac{3\left[1-{\left(1.5\right)}^{\mathrm{n}}\right]}{1-1.5}$

Calculation:

Consider the sequence is,

  ${\mathrm{a}}_{\mathrm{n}}=\frac{3\left[1-{\left(1.5\right)}^{\mathrm{n}}\right]}{1-1.5}$

In the graphing calculator select the mode as shown in Figure 1

  

Figure 1

Input the sequence as shown in Figure 2

  

Figure 2

Adjust the viewing window as shown in Figure 3

  

Figure 3

Press the graph button as shown in Figure 4

  

Figure 4

The above figure shows that the sequence converges.

Consider that the value of $\mathrm{u}\left(\mathrm{n}\right)$ is closer to the finite values as $\mathrm{n}$ increase unboundedly the sequence converges and the limit is,

  $\begin{array}{c}\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }=\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\frac{3\left[1-{\left(0.5\right)}^{\mathrm{n}}\right]}{1-0.5}\\ =\frac{3\left(1-0\right)}{1-0.5}\\ =6\end{array}$