Chapter 7.4, Problem 2E

Solution

To determine : if the series ${\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}}$ has sum or not.

The sum of the series ${\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}}$ exists if $\left|r\right|\ge 1$ .

Given information :

The series ${\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}}$ .

Explanation :

Consider the series ${\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}}$ ,

The sum of infinite geometric series ${\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}}$ with first term $a_1$ and common ration $r$ is calculated using the formula $S=\frac{a_1}{1-r}$ .

Here, $\left|r\right|<1$ . If $\left|r\right|\ge 1$ , then the series has no sum.

For example, consider the series ${\displaystyle \sum _{n=1}^{\infty }2{\left(\frac{1}{5}\right)}^{n-1}}$ .

For the given series the value of $a_1$ is 2 and $r$ is $\frac{1}{5}$ .

As the common ratio $\left|\frac{1}{5}\right|<1$ , the sum of the series exists.

Thus, the sum of the series ${\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}}$ exists if $\left|r\right|\ge 1$ .