Chapter 11.6, Problem 10WE

To determine

To calculate:

Sum of given geometric series.

Answer to Problem 10WE

Sum of given geometric series is $0.66$ .

Explanation of Solution

Give information:

  $4^{-1/2}+4^{-3/2}+4^{-5/2}+4^{-7/2}\mathrm{...}$

Calculation:

In order to find out common ratio of given number,

  $4^{-1/2}+4^{-3/2}+4^{-5/2}+4^{-7/2}\mathrm{...}$

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

  $\begin{array}{l}=\frac{t_2}{t_1}\\ =\frac{4^{-3/2}}{4^{-1/2}}\\ =0.25\end{array}$

Since, $\left|r\right|=\left|0.25\right|<1$

Solution exist.

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.

  $\begin{array}{l}S=\frac{4^{-1/2}}{1-\left(0.25\right)}\\ S=\frac{0.5}{0.75}\\ S=0.66\end{array}$

Therefore, sum of given geometric series is $S=0.66$