Chapter 8.3, Problem 5E

To determine

The values of common ratio $r$ for finding the sum of infinite geometric series.

Answer to Problem 5E

The values of common ratio $r$ should satisfy $\left|r\right|<1$ .

Explanation of Solution

Let $a_1,a_{1}r,a_1{r}^2,\mathrm{........}{a}_1{r}^{n-1},\mathrm{...}$ be an infinite geometric sequence with common ratio $r\ne 1$ and its $n^{th}$ partial sum is given by

  $S_n=a_1+a_{1}r+a_1{r}^2+\mathrm{........}+a_1{r}^{n-1}$

The sum of an infinite geometric series is given by

  $S=\frac{a_1}{1-r}$

So, for the sum being convergent and for finding the sum $r$ should satisfy $\left|r\right|<1$ .