Chapter 12.3, Problem 57AYU

To determine

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

Answer to Problem 57AYU

The series is divergent.

Explanation of Solution

Given information:

The series is $8+12+18+27+\mathrm{........}$ .

Calculation:

For the given series $a_1=8$ , $a_2=12$ and $a_3=18$ .

Calculate $\frac{a_2}{a_1}$ .

  $\begin{array}{c}\frac{a_2}{a_1}=\frac{12}{8}\\ =\frac{3}{2}\end{array}$

Calculate $\frac{a_3}{a_1}$ .

  $\begin{array}{c}\frac{a_3}{a_1}=\frac{18}{12}\\ =\frac{3}{2}\end{array}$

Since the ratio of successive terms is same it is a geometric series with common ratio $r=\frac{3}{2}$ .

Since $\left|r\right|>1$ the series is divergent.

Therefore, the series is divergent.