Worm on the rubber band paradox problem The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the "worm on the rubber band".Suppose that a worm crawls along an infinitely-elastic one-meter rubber band at the same time as the rubber band is uniformly stretched. If the worm travels $1$ centimeter per minute and the band stretches $1$ meter per minute, will the worm ever reach the end of the rubber band? The answer, counterintuitively, is "yes", for after n minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is
$$\dfrac{1}{100}\sum_{k=1}^{n}\dfrac{1}{k}\tag{Wikipedia}$$
Q- It says that worm travels $1$ centimeter per minute and rubber band is stretched by $1$ meter per minute, so after $n^{th}$ minute worm would have covered n centimeters and rubber band would have been stretched by $n$ meters, so at any minute, distance covered by worm would be less than the total length of the rubber band, so how can it ever reach the end of the rubber band? 
 A: The band stretches uniformly, so that means if the worm's present position on the band is some fraction $p$ of the total length of the band, where $0 \le p \le 1$, then the stretching of the band does not change that proportion.  It's equivalent to saying if I took a rubber band and marked the halfway point, and then I stretched the band uniformly to twice its former length, the marked point still represents the midpoint of the stretched band.
Therefore, the only thing that changes the proportion of the worm's progress is its own movement on the band.  Once you understand this, it becomes clear that the worm must complete the traversal so long as it moves at some constant absolute velocity.
The reason why your road analogy doesn't work is because you're assuming the road lengthens from the end, rather than stretching uniformly.  If the worm stays still on the midpoint, stretching the band still keeps the worm at the midpoint because the distance the worm has already traveled also stretches in equal amount to the distance the worm has yet to travel.  Then the worm takes a step, and now the worm is further than halfway along the band.  This is not the case with your road example.
