1. For n, k nonnegative integers with 0 ≤ k ≤ n, recall that the binomial coefficient, (2) can be written as n(n−1)…..(n−k+1). We generalize it as follows k! - r(r — 1) · · · (r — k + 1) - == k k! (1.1) for r € R and k nonnegative integer. Consequently, we have the binomial expansion for rЄ R. (1+x)" = Σ xk k k=0 (1.2) Recall the 1-D simple random walk (SRW): Pi,i+1 = p, Pi,i−1 = q where p + q = 1; and Pi,j = 0 for 0 ≤ |i – j| or |i – j| > 1. Show that 1 (i) (-1/2) = (−1) (2) 2, for n = 0, 1, 2, ...; (ii) Po,o(s) = (1-4pqs²)-1/2 for s <1; (Here Po,o(s) is the generating function of the Pro's where Pro denotes the conditional probability the SRW is at state 0 at time n given that it started at 0.) (iii) Compute lim,→1- Po,o(s) for p 1/2 and p = 1/2. This problem is a continuation of T3Q1 on 1-D Simple Random Walk. Refer to our course notation. (i) Show that Fo,o(s) = 1 - √√1 - 4pqs² for |s| < 1. (ii) Hence, or otherwise, compute for,0 for each p = (0, 1). (iii) For p = 1/2, show that Σn-1 nf,0 = ∞.