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For a geometric series to converge (meaning it approaches a finite value), the common ratio, r, must be between -1 and 1. In our case, with \(r= \frac{1}{2}\), the series converges because \(-1 < r < 1\), thus we can proceed to find the sum of the series. #Step 3: Calculate the Sum of the Infinite Series#

In Achilles and the Tortoise, each time Achilles reaches where the tortoise was, the tortoise will have moved a certain distance further. This can be seen as a fraction of the original distance Achilles had to cover to catch up to the tortoise. As a result, the infinite series describes the sum of distances Achilles covers as he tries to catch up to the tortoise. The fact that the sum of this series converges to 1 demonstrates that despite an infinite number of smaller intervals, Achilles will eventually catch the tortoise once he has covered the full distance. This resolves Zeno's Paradox by providing a mathematical basis for understanding how a seemingly infinite sum of distances can ultimately converge to a finite value.