"Abstractions are fine, but I think people also have to breathe air and eat bread.'' is a quote from Das Glasperlenspiel, or The Glass Bead Game, a novel published in 1946 written by the German author Hermann Hesse. You are playing with n n distinct glass beads B_{1}, B_{2}, ldots, B_{n} B 1 ,B 2 ,…,B n in some order. In each step, you move exactly one bead to the front. The cost of moving the bead to the front is the number of beads before that bead. For example, if we move B_{1} B 1 in the list B_{2}, B_{4}, B_{3}, B_{1} B 2 ,B 4 ,B 3 ,B 1 the total cost is 3 3 and the resulting sequence of beads is B_{1}, B_{2}, B_{4}, B_{3} B 1 ,B 2 ,B 4 ,B 3 . Suppose that at each step glass bead B_{i} B i is moved with probability p_{i}>0 p i >0, where sum_{i=1}^{n} p_{i}=1 ∑ i=1 n p i =1. What is the limit of the expected cost of the m m-th move, when m m tends to infinity?
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