Chapter 11, Problem 44SGR

To determine

Find the sum of the infinite series $\frac{5}{6}-\frac{20}{16}+\frac{80}{54}-\frac{320}{162}+\mathrm{....}$ , if it exists.

Answer to Problem 44SGR

The sum does not exist.

Explanation of Solution

Given:

Find the sum of the infinite series $\frac{5}{6}-\frac{20}{16}+\frac{80}{54}-\frac{320}{162}+\mathrm{....}$ , if it exists.

Concept Used:

  $a_1=\frac{5}{6};r=-\frac{20}{18}÷\frac{5}{6}=-\frac{20}{18}·\frac{6}{5}=-\frac{4}{3}$

If the common ratio r lies between -1 to 1, we can have the sum of an infinite geometric series. That is, the sum exits for  $\left|r\right|<1$

The sum S of an infinite geometric series with  $-1<r<1$ is given by the formula,

  ${\text{S}}_{\infty }=\frac{a_1}{1-r}$  $\text{If}-1<r<1$ , the sum exists.

Sum of infinite GP series: ${\text{S}}_{\infty }=\frac{a_1}{1-r}$

  $\text{If}-1<r<1$ , the sum exists.

Here $r=-\frac{4}{3}\text{and}\left|-\frac{4}{3}\right|>1$ , the sum does not exist.

Thus, the sum does not exist.