Chapter 11, Problem 19MCQ

To determine

Find the sum of the series, ${\displaystyle \sum _{n=1}^{\infty }4}·{(0.5)}^{n-1}$ if it exists.

Answer to Problem 19MCQ

The sum of the infinite series is 8.

Explanation of Solution

Given:

  ${\displaystyle \sum _{n=1}^{\infty }4}·{(0.5)}^{n-1}$

Concept Used:

  ${\displaystyle \sum _{n=1}^{\infty }4}·{(0.5)}^{n-1}$

In the first term n = 1

  $n=1\Rightarrow 4·{(0.5)}^{1-1}=4$

The base of the exponential function is the common ratio r: so r = 0.5

Sum of infinite GP series: ${\text{S}}_{\infty }=\frac{a_1}{1-r}$

  $\text{If}0<r<1$ , the sum exists. Here the common ratio is 0.5, the sum exist.

The common ratio is 0.5 (less than 1), the sum exist.

  $\begin{array}{l}{a}_1=4;r=0.5;\text{find}{\text{S}}_{\infty }\\ {\text{S}}_{\infty }=\frac{a_1}{1-r}=\frac{4}{1-0.5}=\frac{4}{0.5}=8\\ {\text{S}}_{\infty }=8\end{array}$

Thus, the sum of the infinite series is 8.