Chapter 12.3, Problem 65AYU

To determine

To find: Whether the geometric series converges or diverges and find the sum if converges.

Answer to Problem 65AYU

The series is convergent and the sum of series is $6$ .

Explanation of Solution

Given information:

The series is ${\displaystyle \sum _{k=1}^{\infty }3{\left(\frac{2}{3}\right)}^k}$ .

Calculation:

Calculate the first term.

  $\begin{array}{c}{a}_1=3\left(\frac{2}{3}\right)\\ =2\end{array}$

For the given series $a_1=2$ , and $r=\frac{2}{3}$

Since $\left|r\right|<1$ the series is convergent.

The sum of infinite series which is convergent is given by,

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{2}{1-\frac{2}{3}}\\ =\frac{5}{\frac{1}{3}}\\ =6\end{array}$

Therefore, the series is convergent and the sum of series is $6$ .