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The bounces are an infinite series, each with a time attached to them tending to zero.

Time itself is not a series, this is where the fallacy creeps in.

A infinite series can have a finite sum.

True but then in what order are you calculating things. In order to have an infinite series for the air time to sum you must assume there are infinitely many jumps so if there are infinitely many jumps then the ball never rolls on the floor by definition. But then if you say you sum the infinite series of time intervals and the ball stops at that time then you don't have infinitely many bounces because if there were infinitely many bounces the ball would not roll on the floor. So you are missing something somewhere.

http://en.wikipedia.org/wiki/Geometric_series

See also: http://en.wikipedia.org/wiki/Zenos_paradox (Hint: It's not really a paradox.)

This doesn't address the fact you are making a logical fallacy. You can't make a calculation assuming the ball bounces infinitely often and then after the calculation go back and say the ball stops bouncing because it invalidates your original calculation.

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