Chapter 12, Problem 29RE

To determine

To find: The sum of the given expression.

Answer to Problem 29RE

The sum of the expression is $0.49977$ .

Explanation of Solution

Given:

The given expression is,

  ${\displaystyle \sum _{k=1}^7{\left(\frac{1}{3}\right)}^k}$

Calculation:

Consider the given expression is,

  ${\displaystyle \sum _{k=1}^7{\left(\frac{1}{3}\right)}^k}$

Consider the value of first term is,

  $\begin{array}{c}{a}_1={\left(\frac{1}{3}\right)}^1\\ =\frac{1}{3}\end{array}$

Consider the value of second term is,

  $\begin{array}{c}{a}_2={\left(\frac{1}{3}\right)}^2\\ =\frac{1}{9}\end{array}$

Consider the value of third term is,

  $\begin{array}{c}{a}_3={\left(\frac{1}{3}\right)}^3\\ =\frac{1}{27}\end{array}$

Consider the common difference is,

  $\begin{array}{c}\frac{a_3}{a_2}=\frac{\left(\frac{1}{27}\right)}{\left(\frac{1}{9}\right)}\\ =\frac{1}{3}\end{array}$

Consider the formula for the sum of geometric series is,

  $S_n=a_1\left(\frac{1-r^n}{1-r}\right)$

Then,

  $\begin{array}{c}{S}_7=\left(\frac{1}{3}\right)\left(\frac{1-{\left(\frac{1}{3}\right)}^7}{1-\left(\frac{1}{3}\right)}\right)\\ =\left(\frac{1}{2}\right)\left(\frac{2186}{2187}\right)\\ =0.49977\end{array}$