Chapter 11.6, Problem 4ST

a.

To determine

To calculate:

Sum of geometric series if series has no sum, say no.

Answer to Problem 4ST

Given geometric series has sum of $15.6$ .

Explanation of Solution

Give information:

  $\frac{25}{8}+\frac{5}{2}+2+\frac{8}{5}$

Calculation:

In order to find out sum of geometric progression,

  $\frac{25}{8}+\frac{5}{2}+2+\frac{8}{5}$

Common ratio $\left(r\right)$ is calculated as,

  $\begin{array}{l}=\frac{t_2}{t_1}\\ =\frac{\left(5/2\right)}{\left(25/8\right)}\\ =\frac{2.5}{3.12}\\ =0.80\end{array}$

Sum of given geometric series is calculated as,

  $S=\frac{a_1}{1-r}$

Where,

  $a_1=$ Initial term.

Put the values in the equation.

  $\begin{array}{l}S=\frac{25/8}{1-\left(0.80\right)}\\ S=\frac{25/8}{0.2}\\ S=15.6\end{array}$

Therefore, given geometric series has sum of $15.6$ .

b.

To determine

To calculate:

Sum of geometric series if series has no sum, say no.

Answer to Problem 4ST

The finite sum does not exist.

Explanation of Solution

Give information:

  $1+1.1+1.21+-1.331+\mathrm{...}$

Calculation:

In order to find out sum of geometric progression,

  $1+1.1+1.21+-1.331+\mathrm{...}$

Common ratio $\left(r\right)$ is calculated as,

  $\begin{array}{l}=\frac{t_2}{t_1}\\ =\frac{\left(1.1\right)}{\left(1\right)}\\ =1.1\end{array}$

Since $\left|r\right|=\left|1.33\right|>1$

The finite sum does not exist.