I have received a lot answers, which are copied from others. Please don't do that, or I won't accep it!!! Thank you!

Problem 3 (20p). For each j ( Z, let r; ( [0, co) be given. Let (X,)2, be a Markov chain on E :=Zy with transition matrix II : E x E -> [0, 1] given by e if k = j + 2, II(j, k) = if k = 0, otherwise. Suppose that _;12; = co. Determine whether the state 0 is recurrent or transient