Will the worm reach the end of the rubber band?
A worm is at the beginning of an infinitely flexible rubber band which has a length of 1 kilometre.
During a one second interval, the worm advances one centimetre. However, the rubber band's length increases to 2 kilometres. The worm is slightly pulled forward as a consequence of this.
In the following second, the worm advances another centimetre. The rubber band's length increases to 3 kilometres. And so on.
It seems pretty obvious that the worm never reaches the end of the rubber band. However, it does! Why?
In the 1st second the worm covers 1/100,000 of the rubber band's distance. This is 1cm divided by 100,000 cm (1 km).
In the 2nd second it covers 1/200,000, in the 3rd second it covers 1/300,000 , etc.
The sum of the proportions covered is 1/100,000 (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...); will this sum ever reach 1 (100% of the distance). It surprisingly does.
The so-called harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...) is fascinating. It seems that if one keeps summing smaller and smaller numbers, the sum always needs to converge to some finite number. If one sums the first 65,536 terms of the series using Excel, the sum appears to converge to around 11.67. But if we keep adding numbers, by the time we are adding 1/1,000,000 the sum has already increased to 14.39. By the time 1/2,000,000 is added, the sum has kept increasing to 15.09. And it can get as large as we want. The proof is here.
The worm will take more than the estimated age of the Universe to reach the end of the rubber band (which by that stage will be larger than the diameter of the known Universe), but it will eventually get there.
During a one second interval, the worm advances one centimetre. However, the rubber band's length increases to 2 kilometres. The worm is slightly pulled forward as a consequence of this.
In the following second, the worm advances another centimetre. The rubber band's length increases to 3 kilometres. And so on.
It seems pretty obvious that the worm never reaches the end of the rubber band. However, it does! Why?
In the 1st second the worm covers 1/100,000 of the rubber band's distance. This is 1cm divided by 100,000 cm (1 km).
In the 2nd second it covers 1/200,000, in the 3rd second it covers 1/300,000 , etc.
The sum of the proportions covered is 1/100,000 (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...); will this sum ever reach 1 (100% of the distance). It surprisingly does.
The so-called harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...) is fascinating. It seems that if one keeps summing smaller and smaller numbers, the sum always needs to converge to some finite number. If one sums the first 65,536 terms of the series using Excel, the sum appears to converge to around 11.67. But if we keep adding numbers, by the time we are adding 1/1,000,000 the sum has already increased to 14.39. By the time 1/2,000,000 is added, the sum has kept increasing to 15.09. And it can get as large as we want. The proof is here.
The worm will take more than the estimated age of the Universe to reach the end of the rubber band (which by that stage will be larger than the diameter of the known Universe), but it will eventually get there.
posted by DS at 3:41 PM
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