Chapter 9, Problem 57RE

Solution

To identify: Whether the geometric series converges. If it does, find its sum ${\displaystyle \sum _{K=1}^{\infty }2{\left(\frac{3}{4}\right)}^J}$ .

This is convergent series, sum is $6$ .

Given information:

These is given term $r=\frac{3}{4}$ .

Explanation:

Consider the given term.

It is given that $r=\frac{3}{4}$ .

Determine the value of $a_1$

  $\begin{array}{l}{a}_1=2\cdot \frac{3}{4}\\ a_1=\frac{3}{2}\end{array}$

The sum nth term of a geometric sequence with first term and common ratio is given by: $s_n=\frac{a_1}{1-r}$

Put the value of $a_1=\frac{3}{2}\text{and}r=\frac{3}{4}$ into $s_n=\frac{a_1}{1-r}$ .

  $\begin{array}{l}s=\frac{a_1}{1-r}\\ s=\frac{\frac{3}{2}}{1-\frac{3}{4}}\\ s=\frac{\frac{3}{2}}{\frac{1}{4}}\\ s=\frac{3}{2}\cdot \frac{4}{1}\\ s=6\end{array}$

So, $\left|r\right|=\left|\frac{3}{2}\right|<1$ ,the series converges.