Chapter 12.3, Problem 53AYU

To determine

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

Answer to Problem 53AYU

The series is convergent and the sum of series is $16$ .

Explanation of Solution

Given information:

The series is $8+4+2+\mathrm{.........}$ .

Calculation:

For the given series $a_1=8$ , $a_2=4$ and $a_3=2$ .

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

  $\begin{array}{c}\frac{a_2}{a_1}=\frac{4}{8}\\ =\frac{1}{2}\end{array}$

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

  $\begin{array}{c}\frac{a_3}{a_1}=\frac{2}{4}\\ =\frac{1}{2}\end{array}$

Since the ratio of successive terms is same it is a geometric series with common ratio $r=\frac{1}{2}$ .

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

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

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{8}{1-\frac{1}{2}}\\ =\frac{8}{\frac{1}{2}}\\ =16\end{array}$

Therefore, the series is convergent and the sum of series is $16$ .