Question

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

$8+4+2+....$

Short answer

The given infinite geometric series $8+4+2+....$is convergent and its sum is 16.

Step-by-step solution

Step 1. Write the given information.

The given geometric series is:

$8+4+2+....$

Step 2. Find the common ratio.

$a_1=8,a_2=4,a_3=2$

The common ratio is the ratio of successive terms:

$$\begin{gathered} \frac{4}{8}=\frac{1}{2},\frac{2}{4}=\frac{1}{2} \\ r=\frac{1}{2} \end{gathered}$$

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

If $\left|r\right|<1$then the infinite geometric series $\sum _{k=1}^{\infty }{a}_1{r}^{k-1}$converges.

As we can see that $r=\frac{1}{2}\mathrm{and}\frac{1}{2}<1$, therefore the given series is convergent.

Step 4. Find the sum of the series.

Use the formula to find the sum of the given geometric series $\sum _{k=1}^{\infty }{a}_1{r}^{k-1}=\frac{a_1}{1-r}$,

$$\begin{gathered} \sum _{k=1}^{\infty }8{\left(\frac{1}{2}\right)}^{k-1}=\frac{8}{1-{\displaystyle \frac{1}{2}}} \\ =\frac{8}{{\displaystyle \frac{1}{2}}} \\ =16 \end{gathered}$$

Therefore, the sum of the infinite geometric series is 16.