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## Homework Statement

Give the value of u_0.

## Homework Equations

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

## The Attempt at a Solution

So, we have u_0 = P(X_n = 0 for some n ≥ 0|X_0 = 0) = sum_{n=-infinity}^{infinity}(2n choose n) (pq)^n. Is this on the right track? If so, do I need to go further? How would I? Thanks.

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