Inchworm on a Rubber Rope

Imagine a rubber rope one meter long. An inchworm starts at one end and travels along the rope at 1 cm/sec. At the end of every second, the rope gets stretched so that it was one meter longer than before (the worm is carried along with the stretching). So the worm travels 1 cm, the rope gets stretched 1 whole meter, then the worm travels 1 cm farther on the stretched rope, the rope gets stretched again by another meter, and the worm travels 1 cm farther, etc.

Does the worm ever reach the end of the rope? Amazingly, yes! If you consider the fraction of the rope traveled by the Nth second, the fraction is proportional to the partial sum of the  out to the Nth term. Since this diverges as N grows, it must eventually exceed the total length of rope.

But don’t hold your breath—the number of seconds that it takes the worm to reach the end is longer than the lifetime of the known universe!

Presentation Suggestions:
Draw a suggestive picture. Solving for the amount of time required to reach the end of the rope is also a fun puzzle.

The Math Behind the Fact:
The diverges, but very very slowly! In fact the sum of the first N terms of the harmonic series grows like the natural logarithm of N.

How to Cite this Page: 
Su, Francis E., et al. “Inchworm on a Rubber Rope.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
M. Gardner

Fun Fact suggested by:
Francis Su

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