Chapter 12.3, Problem 2AYU

To determine

To find: The suitable parameter for given blanks.

Answer to Problem 2AYU

The suitable parameter for given blank is ‘ $\frac{a_1}{\left(1-r\right)}$ ’.

Explanation of Solution

Given information:

If $\left|r\right|<1$ ,the sum of geometric series ${\displaystyle \sum _{k=1}^{\infty }a{r}^{k-1}}$ is ________.

Calculation:

The sum of first $n$ terms $S_n$ of a geometric sequence is given by the formula.

  $\begin{array}{l}{S}_n=\frac{a_1\left(1-r^n\right)}{\left(1-r\right)}\\ =\frac{a_1}{\left(1-r\right)}-\frac{a_1{r}^n}{\left(1-r\right)}\end{array}$ ….. $\left(1\right)$ .

If $\left|r\right|<1$ then $\left|r^n\right|$ tends to $0$ as $n\to \infty$ .

Then from equation $\left(1\right)$ term $\frac{a_1{r}^n}{\left(1-r\right)}$ tends to $0$ .

Equation $\left(1\right)$ becomes .

  $S_n=\frac{a_1}{\left(1-r\right)}$

Therefore the blank can be filled as follows.

If $\left|r\right|<1$ ,the sum of geometric series ${\displaystyle \sum _{k=1}^{\infty }a{r}^{k-1}}$ is $\frac{a_1}{\left(1-r\right)}$ .