Chapter 12.3, Problem 55AYU

To determine

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

Answer to Problem 55AYU

The series is convergent and the sum of series is $\frac{8}{5}$ .

Explanation of Solution

Given information:

The series is $2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\mathrm{.......}$ .

Calculation:

For the given series $a_1=2$ , $a_2=\frac{-1}{2}$ and $a_3=\frac{1}{8}$ .

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

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

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

  $\begin{array}{c}\frac{a_3}{a_1}=\frac{\frac{1}{8}}{-\frac{1}{2}}\\ =-\frac{1}{4}\end{array}$

Since the ratio of successive terms is same it is a geometric series with common ratio $r=-\frac{1}{4}$ .

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

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

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{2}{1-\left(-\frac{1}{4}\right)}\\ =\frac{2}{\frac{5}{4}}\\ =\frac{8}{5}\end{array}$

Therefore, the series is convergent and the sum of series is $\frac{8}{5}$ .