Question

"A runner can never reach a finish line 1 mile away because first he would have to run half a mile, and then must run half of the remaining distance, or \(\frac{1}{4}\) mile, and then half of that, and so on. Since there are an infinite number of distances that must be run, it will take an infinite length of time, and so the runner will never reach the finish line." Show that the distances to be run form the infinite series; $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots $$ Then disprove the paradox by actually finding the sum of that series and showing that the sum is not infinite.

Short answer

The sum of the infinite geometric series \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/n, \frac(1/2)/н продолжает идти}though the number of terms is infinite the total sum converges to 1 and is thus finite.

Step-by-step solution

Identify the Pattern

To find the sum of an infinite geometric series, we use the formula $$S = \frac{a}{1 - r}$$ where 'a' is the first term and 'r' is the common ratio. For this series, 'a' is \frac{1}{2} and 'r' is \frac{1}{2}. Applying these to the formula, we get: $$S = \frac{\frac{1}{2}}{1 - \frac{1}{2}}$$

Perform the Calculation

Carrying out the calculation we're left with: $$S = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$$ This means the sum of this infinite series is actually 1, disproving the idea that it would take an infinite length of time to reach the finish line.

Conclusion

The paradox arises from a misunderstanding of infinite series and their sums. While the number of sub-distances the runner must travel is infinite, the sum of these distances is finite and equal to 1 mile, the total distance to the finish line. Therefore, the runner will indeed reach the finish line.

Key concepts

Convergence of Series

In the realm of mathematics, especially when dealing with series, the concept of convergence is fundamental. A series is simply a sum of a sequence of terms.

However, not all series approach a finite value. When we say a series converges, we mean that as you add more and more terms, the sum gets closer and closer to a specific number, rather than growing indefinitely. This specific number is known as the limit of the series.

There's a special kind of series where you can easily determine if it converges or not, known as a geometric series. It consists of terms that each are a constant multiple (the common ratio) of the previous term. To check the convergence of a geometric series, you look at the absolute value of the common ratio. If it is less than one, the series converges; if it is equal to or greater than one, the series diverges, meaning it doesn’t approach a finite limit.

For instance, in the runner's paradox, the series of distances is given by \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\). Here, each term is half the previous term, making the common ratio \(r = \frac{1}{2}\). Since \(r\) is less than 1, this infinite geometric series converges to a finite number. The formula \(S = \frac{a}{1 - r}\) provides us a way to calculate this limit.

Geometric Progression

The core of the runner's paradox revolves around the concept of a geometric progression, or geometric series. This is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

In mathematical terms, if the first term of the series is \(a\), and the common ratio is \(r\), then the sequence is \(a, ar, ar^2, ar^3, \ldots\). The ratio between successive terms is always the same, hence the name 'geometric progression'.

In the example provided, the first term \(a = \frac{1}{2}\) and the common ratio \(r = \frac{1}{2}\). Plugging these into the formula for the sum of an infinite geometric series, \(S = \frac{a}{1 - r}\), demonstrates that the sum of these portions of the run is finite. Concretely, even though the runner seems to have to run an infinite number of distances, as they get progressively smaller, the actual distance that needs to be covered remains usable and is ultimately equal to one mile.

Mathematical Paradoxes

Mathematical paradoxes such as the runner's paradox can be quite intriguing and often arise from either a misinterpretation or misapplication of mathematical principles. A paradox presents a situation that defies intuition or appears to contradict itself.

In the case of the runner's paradox, it seems counterintuitive to think that an infinite number of steps could result in a finite distance. The paradoxical element lies in the distinction between the 'infinite' number of steps and the 'sum' of these steps.

This paradox is addressed by understanding the true nature of infinite geometric series and the principle of convergence. As shown in the solution, by applying the correct formula, one can see that the total distance converges on a finite limit, hence debunking the paradox.

Paradoxes serve as important tools for learning because they challenge our understanding of concepts, prompting deeper inquiry and discovery. They push the boundaries of known mathematics and often lead to new insights or the clarification of existing concepts.