Examples of problems that are easier in the infinite case than in the finite case. I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.

A Markov chain is characterized by a discrete probability distribution of the initial state of a system, 0, and a transition matrix P such that pij is the probability of going from state i to state j after any one transition. Under certain assumptions, the probability distribution of states after the first transition is 1 = 0P, after two transitions is 2 = 1P = 0P2, after n transitions, n = 0Pn.  
By constrast, the steady-state (as n -> ∞) probability distribution of states, , must satisfy  = P, and  = 1, where  is a column vector of 1's. This is an over-determined (by one) system of equations, but the upshot is that the steady-state probability distribution of states is obtained by inverting a matrix.
See, for example, the Markov chain Wiki for details.
