Chapter 11.6, Problem 9WE

To determine

To calculate:

Sum of given geometric series.

Answer to Problem 9WE

Sum of given geometric series is $0.3$

Explanation of Solution

Give information:

  $3\sqrt{3}-3+\sqrt{3}-1+\mathrm{...}$

Calculation:

In order to find out common ratio of given number,

  $3\sqrt{3}-3+\sqrt{3}-1+\mathrm{...}$

Here first term is $a_1=3\sqrt{3}$

So common ratio is $r=\frac{-3}{3\sqrt{3}}=\frac{-3}{5.19}=-0.57$

Since $\left|r\right|=\left|-0.57\right|<1$

Infinite sum exists.

So, Sum of given geometric series is calculated as,

  $=\frac{a^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}=\frac{a^n}{1-r}\\ =\frac{3\sqrt{3}}{1-\left(-0.57\right)}\\ =\frac{3\sqrt{3}}{1+\left(0.57\right)}\\ =\frac{3\sqrt{3}}{1.57}\\ =\frac{5.19}{1.57}\\ =3.30\end{array}$

Therefore, sum of given geometric series is $3.30$ .