Chapter 9, Problem 3PS

To determine

To show: that both series have finite sums, and to find these sums represent.

Answer to Problem 3PS

The sum of the distance and time series are 40 feet and 2 seconds respectively.

Explanation of Solution

Given information:

Can the Greek hero Achilles, running at 20 feet per second, ever catch a tortoise, starting 20 feet ahead of A and running at 10 feet per second? The Greek mathematician Z said no. When A runs 20 feet, the tortoise will be 10 feet ahead. Then , when A runs 10 feet, the tortoise will be 5 feet ahead. A will keep cutting the distance in half but will never catch the tortoise. The table shows Zeno’s reasoning.

Calculation:

Since the distance form an infinite geometric series with common ratio

  $\begin{array}{c}r=\frac{10}{20}\\ =\frac{1}{2}\\ <1\end{array}$

Therefore, the sum of the distance series will be

  $\begin{array}{c}{S}_1=\frac{a}{1-r}\\ =\frac{20}{1-\frac{1}{2}}\\ =20\left(2\right)\\ =40\end{array}$

Since the time required form an infinite geometric series with common ratio

  $\begin{array}{c}r=\frac{0.5}{1}\\ =0.5\\ <1\end{array}$

Therefore, the sum of the distance series will be

  $\begin{array}{c}{S}_2=\frac{a}{1-r}\\ =\frac{1}{1-0.5}\\ =2\end{array}$

Thus, the sum of the distance and time series are 40 feet and 2 seconds respectively

Hence, both the series have finite sums.