Chapter 1.1, Problem 12P

In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth 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.