Chapter 12, Problem 16RE

To determine

To find:The given sum.

Answer to Problem 16RE

The given sumis $0.49977$ .

Explanation of Solution

Given information:

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

Calculation:

The sum is,

  $\begin{array}{c}{\displaystyle \sum _{k=1}^7{\left(\frac{1}{3}\right)}^k}=\frac{1}{3}\left[\frac{1-{\left(\frac{1}{3}\right)}^7}{1-\left(\frac{1}{3}\right)}\right]\\ =\frac{1}{3}\left[\frac{1-{\left(\frac{1}{3}\right)}^7}{\frac{2}{3}}\right]\\ =\frac{1}{2}\left[1-{\left(\frac{1}{3}\right)}^7\right]\\ =\frac{1093}{2187}\\ =0.49977\end{array}$