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Zeno's Paradox

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Ancient mathematicians did not have a theory of limits. A Greek philosopher named Zeno of Elea (5th century B.C) devised the following paradox.

Achilles and the tortoise had a race. Achilles could run 10 times as fast as the tortoise, but the tortoise had a hundred yard start. Achilles runs the hundred yards, but the tortoise is now 10 yards ahead. Achilles runs the 10 yards, but the tortoise is now 1 yard ahead. Achilles runs the 1 yard, but the tortoise is now 1/10 yard ahead, and so on. How can Achilles overtake the tortoise?

Because the ancient Greeks did understand limits, in their logic the problem could not be solved. However, we know that the sum of the series

\sum_{i \mathop =0}^{\infty} 100 (\frac{1}{10})^i = 100 + 10 + 1 + \frac{1}{10} + \frac{1}{100}... = \frac{1000}{9} = 111.111...

At this point, Achilles overtakes the tortoise.

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