[greenlblue]

Ya, and what are the terms in the series representing? Is it air time of each bounce? Your calculation makes it clear that it is. So you are saying you are calculating the air time for infinitely many bounces, the key word here is infinite, i.e. the ball bounces up and down, up and down infinitely often. So if the ball bounces infinitely often how can it stop and roll on the floor because infinitely many bounces means not stopping after finitely many bounces and rolling on the floor. Your calculation for the time is confounding two things, air time and bouncing. You can calculate the air time assuming infinitely many bounces but then you can't go and claim that the ball stops bouncing after t = whatever because you calculated t = whatever assuming the ball never stopped bouncing and then after the calculation went back and changed your assumption, that is a logical fallacy if I ever saw one.

[mkarmac]

1) The ball stops bouncing in a finite amount of time.

2) The ball bounces an infinite number of times.

3) This is not a contradiction, no more than the idea that a projectile passes through an infinite number of spatial points in finite time. Please go and read about geometric series and Zeno's paradox on wikipedia, as another commenter has already suggested.