Bouncing Balls

A bouncing ball, time zero as the start of the first bounce, is described by a fixed ratio r < 1 from one height for n = k to the next, as the height for n = k + 1 so that each is a bit lower. A ball makes infinitely many bounces but in a finite time, what is the formula for the total time T to standstill?

I haven't done the calculations yet but I do know how, will give the formula in this forum entry soon. If someone is not before me! I am writing an ABC story also.

Sketches

https://www.abctales.com/sites/abctales.com/files/u52179/illustration%20...