Let p>q>0 with p+q = 1 and a = q/p < 1.

Question

Let X_n denote the random walk with transitions
X_{n+1} = CASE 1: X_n + 1 with probability p and CASE 2: X_n - 1 with probability q.

For i ≥ 0, we set u_i = P(X_n = 0 for some n ≥ 0|X_0 = i).

Give the value of u_0.