bug on band
An infinitely stretchable rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s; initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?
Solution by Michael A. Gottlieb
I imagine the bug crawling 0.001 centimeter then the band being stretched 1 meter alternately, in steps 1 second apart. (So, this solution will only be approximate.) The bugs progress (distance from the wall, in meters) at the nth step p[n] is then given by the recursive relation:
p = .00001 m
p[n] = p[n-1] * n/(n-1) + .00001 for n>1
Expanding this, I find:
p = .00001
p = (.00001 * 2/1 + .00001 = .00001 * (2/1 + 2/2) = .00001 * 2 * (1 + 1/2)
p = .00001 * (2/1 + 2/2) * 3/2 + .00001 = .00001 * (3/1 + 3/2 + 3/3) = .00001 * 3 * (1 + 1/2 + 1/3)
p[n] = .00001 * n * (Sum from i=1 to n of 1/i)
For large n the Sum is approximately equal to log n. Since at step n the rubber band is n meters long, the bug reaches the end of the rubber band when
.00001 n log n = n.
Thus the bug reaches the end of the band in approximately e^100000 seconds.