Chapter 7.5, Problem 9MRPS

Solution

To state: An example of an infinite geometric series that has a sum of 4.

The infinite geometric series is ${\displaystyle \sum _{n=0}^{\infty }2\cdot {\left(0.5\right)}^n}$ .

Given information:

The given statement says to an example of an infinite geometric series that has a sum of 4.

Explanation:

The sum of an infinite geometric series with first term $a_1$ and common ratio r when r is $r<1$ is:

  $S=\frac{a_1}{1-r}$

Now the given sum is 4. Then:

  $4=\frac{a_1}{1-r}$

Let’s assume the value of r as 0.5. Now substitute the value of r ,

  $\begin{array}{l}4=\frac{a_1}{1-0.5}\\ 4=\frac{a_1}{0.5}\\ 2=a_1\end{array}$

Therefore, the infinite geometric series is:

  ${\displaystyle \sum _{n=0}^{\infty }{a}_1\cdot r^n}={\displaystyle \sum _{n=0}^{\infty }2\cdot {\left(0.5\right)}^n}$