Chapter 8.1, Problem 109E

To determine

To find: The sum third, fourth and fifth sum of the series.

Answer to Problem 109E

The third partial sum of the given series is $0.875$ .

The fourth partial sum of the given series is $0.9375$ .

The fifth partial sum of the given series is $0.96875$ .

Explanation of Solution

Given information:$$

  ${\displaystyle \sum _{i=1}^{\infty }{\left(\frac{1}{2}\right)}^i}$

Calculation:

Finding third partial sum of the series,

  $\begin{array}{l}{\displaystyle \sum _{i=1}^3{\left(\frac{1}{2}\right)}^i={\left(\frac{1}{2}\right)}^1+{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^3}\\ {\displaystyle \sum _{i=1}^3{\left(\frac{1}{2}\right)}^i}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\\ {\displaystyle \sum _{i=1}^3{\left(\frac{1}{2}\right)}^i}=0.5+0.25+0.125\\ {\displaystyle \sum _{i=1}^3{\left(\frac{1}{2}\right)}^i}=0.875\end{array}$

Finding fourth partial sum of the series,

  $\begin{array}{l}{\displaystyle \sum _{i=1}^4{\left(\frac{1}{2}\right)}^i={\left(\frac{1}{2}\right)}^1+{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^4}\\ {\displaystyle \sum _{i=1}^4{\left(\frac{1}{2}\right)}^i}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}\\ {\displaystyle \sum _{i=1}^4{\left(\frac{1}{2}\right)}^i}=0.5+0.25+0.125+0.9375\\ {\displaystyle \sum _{i=1}^4{\left(\frac{1}{2}\right)}^i}=0.9375\end{array}$

Finding fifth partial sum of the series,

  $\begin{array}{l}{\displaystyle \sum _{i=1}^5{\left(\frac{1}{2}\right)}^i={\left(\frac{1}{2}\right)}^1+{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{2}\right)}^4+{\left(\frac{1}{2}\right)}^5}\\ {\displaystyle \sum _{i=1}^5{\left(\frac{1}{2}\right)}^i}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}\\ {\displaystyle \sum _{i=1}^5{\left(\frac{1}{2}\right)}^i}=0.5+0.25+0.125+0.0625+0.03125\\ {\displaystyle \sum _{i=1}^5{\left(\frac{1}{2}\right)}^i}=0.96875\end{array}$

Thus we can find the third, fourth and fifth partial sum of the given series.