Suppose (a large number of particles are bouncing back and forth between x = 0 and x = 1, except that at each endpoint some escape. Let r be the fraction reflected each time; then (1 - r) is the fraction escaping. Suppose the particles start at x = 0 heading toward x = 1; eventually all particles will escape. Write an infinite series for the fraction which escape at x = 1 and similarly for the fraction which escape at x = 0. Sum both the series. What is the largest fraction of the particles which can escape at x = 0? (Remember that r must be between 0 and 1.)