Mathematical Methods in the Physical Sciences · Problem 1.12

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · geometric-series

Problem

In a water purification process, one-

n

th of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-

n

th of that removed in the preceding stage. Show that if

n = 2

, the water can be made as pure as you like, but that if

n = 3

, at least one-half of the impurity will remain no matter how many stages are used.