Chapter 11.6, Problem 7OE

To determine

To calculate:

Common ratio of given geometric series and sum series has.

Answer to Problem 7OE

Common ratio of given geometric series is $\frac{4}{5}$ and given geometric series has no sum.

Explanation of Solution

Give information:

  ${\displaystyle \sum _{n=0}^{\infty }3{\left(\frac{4}{5}\right)}^n}$

Calculation:

In order to find out common ratio of given number,

  ${\displaystyle \sum _{n=0}^{\infty }3{\left(\frac{4}{5}\right)}^n}$

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

  $r=\frac{4}{5}$

Sum of given geometric series is calculated as,

  ${\displaystyle \sum _{k=0}^{n-1}\left(a{r}^k\right)}=a\frac{1-r^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}{\displaystyle \sum _{k=0}^{n-1}\left(a{r}^k\right)}\\ =a\frac{1-r^n}{1-r}\\ =3\frac{1-{\left(4/5\right)}^0}{1-\left(4/5\right)}\\ =0\end{array}$

Therefore, Common ratio of given geometric series is $\frac{4}{5}$ and given geometric series has no sum.