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I refer you to Zeno's paradox for an example of how a geometric series can allow an infinite number of things to happen in a finite time.

In this case the time taken forms a geometric series, and the total time taken is the sum of that geometric series. Which means that, for the same mathematical reasons that let Achilles catch the tortoise, it stops in finite time.

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I am familiar with Zeno's paradox and all other things Zeno and infinite series summing to finite things but there is fallacy here that nobody seems to get. Yes the time taken is indeed a geometric series but if the time taken is a geometric series, an infinite one at that, then that means there are infinitely many bounces, no? So if there are infinitely many bounces how can you claim the ball stops bouncing? You are confounding two things, air time which is indeed finite because it forms a geometric series, and the number of bounces. You can not have an infinite geometric series to calculate the time and only have finitely many bounces.

The ball bounces infinitely many times, in a finite period of time, and then stops! :) You can model this completely, i.e. describe height as a function of time. You can know exactly where the ball is at any particular time, and it follows a continuous curve. So it does make sense as a model, although it is perhaps a bit mind-bending.

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