Quote:
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Originally Posted by Chuckles
Long Walk to Forever ?*
While studying limits, a friend of yours, who fancies himself as a modern day Zeno**, proposes the following variation on Achilles and the Tortoise story.
Imagine a three-foot elastic band with a small tortoise sitting on one end (fixed) and Achilles, with a delicious flower, holding the other end of the band. Naturally, the hungry tortoise starts to walk toward the flower. However, when the tortoise reaches the one-foot mark, Achilles stretches the (whole) band an additional three feet in length. Undaunted, and perhaps a little slow, the tortoise walks another foot and once again Achilles stretches the band another three feet. If this situation continues stretching in this same manner, will the tenacious tortoise ever reach the end of the band and receive the flower from Achilles?
Find the distance the tortoise walks, and the ratio of the distance walked to the total length of the band.
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Ater discussing (for fun) this problem with some friends, the answer is obvious.
Because the tortoise gets moved along the elastic band as it stretches, this situation will be easier to analyse if we merely measure the
proportion of distance the tortoise has travelled across the elastic band, rather than its literal distance.
So, in the first iteration, the tortoise moves one foot across the three foot distance of the elastic band, making it one third of the way across. Then, Zeno stretches the band an extra three feet. However, this doesn't change the proportion of distance the tortoise has travelled, one thid. It then travels another foot across the, now, 3 + 3 = 6 foot elastic. In other words, it has travelled another 1/6 of the distance. After that, it will travel another foot across a 3 + 3 + 3 = 9 foot elastic band, making it another 1/9 the distance closer. As you can see, the length of the elastic band is
3n at each iteration
n of the sequence, so the proportion of distance travelled will, thus, be
1/(3n). Sum this sequence and it becomes clear that this is 1/3 of the
Hormonic series.
Because the Harmonic series diverges, we know that the tortoise
will get to the flower and that it will do so the same number of times it will take the series to sum to three. Unfortunately, there's no simple formula to represent the sum of the Harmonic series, although it can be approximated by ln(n). This approximation is not very good for a sum as low as three but if the problem were just a little different, like stretching the elastic band twenty feet, instead of just three, then the number of iterations might be pretty close to e<sup>20</sup>...