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Rubber Rope Problem

Anyone up for a mind-expanding problem to work? I like these because they keep my mind sharp. If so, here goes?

A worm is at one end of a rubber rope that can be stretched indefinitely. For the sake of this problem, the worm can live forever and is, for all intents and purposes, dimensionless. At the beginning of the problem, the rope is one kilometer in length. The worm travels from one end of the rope to the other at the rate of one centimeter per second. In other words, each second that elapses, the worm travels a distance of one centimeter. Thus, after the first second, the worm has traveled a distance of one centimeter and the rope is stretched instantly to a total length of two kilometers. After two seconds, the worm has traveled two centimeters, and the rope is stretched instantly until it is now three kilometers in length, and so on. The stretching of the rope is uniform and only the rope is stretched. The rate of speed of the worm and the unit of time is uniform, as well. Remember, the worm never dies and the rope can be stretched indefinitely.

The question is this: Will the worm ever reach the other end of the rope? And, if so, how long will it take? I will publish the answer to this problem in an update to this post at a later time, which will give you time to think about it and hopefully solve it. Good luck!

Keep watching for the solution.