Chapter 9.1, Problem 39E

a.

To determine

To find the geometric series that converges.

Answer to Problem 39E

The geometric series which converges is ${\displaystyle \sum _{n=0}^{\infty }2{\left(\frac{3}{5}\right)}^{n-1}}$ .

Explanation of Solution

Given:

The given series $S={\displaystyle \sum _{n=0}^{\infty }a{r}^{n-1}x}$ and $a=2$ .

Formula used:

If the sequence of partial sum has a limit as $n\to \infty$ , the series converges otherwise the sequence is divergence. For series to be convergence the term should approach to zero. That is common ratio should be less than $1$ . That is $\left|r\right|<1$ , where $r$ is the common ratio.

Calculation:

Given that the geometric series converges.

The initial value is $a=2$

The common ratio is r.

Given that sum of the series is $5$ .

  $\begin{array}{c}S=\frac{2}{1-r}=5\\ \frac{2}{1-r}=5\\ 2=5-5r\\ 5r=3\\ r=\frac{5}{3}\end{array}$

Series obtained is ${\displaystyle \sum _{n=0}^{\infty }2{\left(\frac{3}{5}\right)}^{n-1}}$ .

b.

To determine

To find the geometric series that converges.

  ${\displaystyle \sum _{n=0}^{\infty }\frac{13}{2}{\left(-\frac{3}{10}\right)}^{n-1}}$

Answer to Problem 39E

  ${\displaystyle \sum _{n=0}^{\infty }\frac{13}{2}{\left(-\frac{3}{10}\right)}^{n-1}}$

Explanation of Solution

Given:

The given series $S={\displaystyle \sum _{n=0}^{\infty }a{r}^{n-1}x}$ and $a=\frac{13}{2}$

Formula used:

If the sequence of partial sum has a limit as $n\to \infty$ , the series converges otherwise the sequence is divergence. For series to be convergence the term should approach to zero. That is common ratio should be less than $1$ . That is $\left|r\right|<1$ , where $r$ is the common ratio.

Calculation:

Given that the geometric series converges.

The initial value is $a=\frac{13}{2}$

The common ratio is r.

Given that sum of the series is $5$ .

  $\begin{array}{l}S=\frac{\frac{13}{2}}{1-r}=5\\ \frac{13}{2(1-r)}=5\\ 5r=\frac{3}{2}\\ r=\frac{3}{10}\end{array}$

Series obtained is ${\displaystyle \sum _{n=0}^{\infty }\frac{13}{2}{\left(-\frac{3}{10}\right)}^{n-1}}$.