Question

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

$8+12+18+27+....$

Short answer

The given infinite geometric series $8+12+18+27+....$is diverges.

Step-by-step solution

Step 1. Write the given information.

The given geometric series is:

$8+12+18+27+....$

Find the common ratio.

$a_1=8,a_2=12,a_3=18,a_4=27$

The common ratio is the ratio of successive terms:

$$\begin{gathered} \frac{12}{8}=\frac{3}{2},\frac{18}{12}=\frac{3}{2} \\ r=\frac{3}{2} \end{gathered}$$

Step 3. Determine whether the infinite geometric series is converges or diverges.

If $\left|r\right|<1$then the infinite geometric series converges.

As we can see that $r=\frac{3}{2}\mathrm{and}\frac{3}{2}>1$, therefore the given series is not convergent.

The given infinite geometric series is diverges.