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During the first second, the ant traverses 1/1000 of the band. When the band stretches, he's still 1/1000 of the way to the end. During the second second, the band is 2 meters long, so he only traverses 1/2000 of the band. In general, during the nth second the ant traverses 1/(1000*n) of the rubber band. So, for the time up to and through the nth second, he has traversed:
1/1000 + 1/2000 + 1/3000 + ... + 1/(1000*n)
As n grows large, this series increases without bound. When the value of this series reaches 1, the ant will be at the far end of the rubber band. Therefore, mathematically speaking, the ant will reach the end.
The series above takes as many terms to reach one as the following series takes to reach 1000:
1/1 + 1/2 + 1/3 + ...
This takes between e^999 and e^1000 terms. So, the ant takes roughly e^1000 seconds to reach the end.
But wait. e^1000 seconds is inconcievably huge, roughly 10^426 years. By way of comparison, the universe is only about 10^10 years old, and is expected to suffer complete heat death before 10^100 years.
So yes, the ant gets there, but it takes him 10^326 universe lifetimes to do so. At the end of this time, the rubber band is 10^426 meters, or 10^418 light-years long. Assuming that the rubber band weighs 10 grams (reasonable for a 1 meter band), it has about 10^24 atoms, and the mean distance between atoms when fully stretched is 10^394 light-years. The known size of the universe is only about 10^10 light-years, so the rubber band is stretched much further than that.
All of which goes to show that, mathematics notwithstanding, the ant will never reach the end. |
