Infinity

Apr 25, 2026 - 00:47

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Zeno: Achilles and the tortoise

← Previous revision		Revision as of 19:16, 24 April 2026
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	Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with {{math\|1=''a'' = 10 seconds}} and {{math\|1=''x'' = 0.01}}. Achilles does overtake the tortoise; it takes him		Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with {{math\|1=''a'' = 10 seconds}} and {{math\|1=''x'' = 0.01}}. Achilles does overtake the tortoise; it takes him
	: $10+0.1+0.001+0.00001+\cdots$ $=\frac {10}{1-0.01}= \frac {10}{0.99}=10.10101\ldots\text{ seconds}.$		: $10+0.1+0.001+0.00001+\cdots$ $=\frac {10}{1-0.01}= \frac {10}{0.99}=10.10101\ldots\text{ seconds}.$

			the paradox can be represented by
			$\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=?$
			whereas the sum is later found to be equal to exactly 1

	===Early Indian===		===Early Indian===