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?
How to write this power series problem in mathematical notation?
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power-series
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0i assume the amount of particles = a, so at x = 1 the amount of particles escapes = a(1-r), the second bounce = a(1-r)^2, . . ., i'm using the S = a/ 1 − r formula and stuck at summing both the series, – 2017-02-19
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0Each separate series is geometric with ratio $r^2$, so use the same formula. – 2017-02-19