Chapter 12.3, Problem 4E

(a)

To determine

To fill: The statement “The nth partial sum of a geometric sequence $a_n=a{r}^{n-1}$ is given by ________.”

Answer to Problem 4E

The nth partial sum of a geometric sequence $a_n=a{r}^{n-1}$ is given by $S_n=a\cdot \frac{1-r^n}{1-r}$.

Explanation of Solution

The terms of the geometric sequence are, $a,ar,a{r}^2,a{r}^3,\cdots$ and its nth term is $a_n=a{r}^{n-1}$

Partial sum of the geometric sequence $a+ar+a{r}^2+a{r}^3+\cdots +a{r}^{n-1},r\ne 1$ is, $S_n=a\cdot \frac{1-r^n}{1-r}$.

(b)

To determine

To fill: The statement “The series ${\displaystyle \sum _{k=1}^{\infty }a{r}^{k-1}}=a+ar+a{r}^2+a{r}^3+\dots$ is an infinite _______ series. If $\left|r\right|<1$, then this series _________, and its sum is ________.”

Answer to Problem 4E

The series ${\displaystyle \sum _{k=1}^{\infty }a{r}^{k-1}}=a+ar+a{r}^2+a{r}^3+\dots$ is an infinite geometric series. If $\left|r\right|<1$, then this series converges, and its sum is $S=\frac{a}{1-r}$.

Explanation of Solution

The terms of the geometric sequence are, $a,ar,a{r}^2,a{r}^3,\cdots$

Thus, the series ${\displaystyle \sum _{k=1}^{\infty }a{r}^{k-1}}=a+ar+a{r}^2+a{r}^3+\dots$ is a geometric series. So, it is said to be infinite geometric series as k ranges from 1 to infinity.

If the absolute value of common ratio is less than 1 then this series converges and it is calculated by the formula, $S=\frac{a}{1-r}$.