- #1
nonequilibrium
- 1,439
- 2
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I'll shortly explain what my reasoning is so far, but please ignore it if it comes across too jumbled:
---
Let P denote the markov matrix associated with this problem, then I think I was able to argue that the probability that is asked for is equal to [itex]1- \sum_{n=1}^{+\infty} P^n(0,0)[/itex] where [itex]P^n(0,0)[/itex] denotes the element in the first row of the first column of the n-th power of the Markov matrix.
And I then wanted to calculate [itex]P^n(0,0)[/itex] for every n by trying to find a pattern in [itex]P^1(0,0)[/itex], [itex]P^2(0,0)[/itex], [itex]P^3(0,0)[/itex], etc. I think I found one: define [itex]t_n = \left(t_{n-1} + \left( \frac{n+1}{n} \right)^2 \prod_{k=1}^{n+1} x_k \right) x_n[/itex] (with [itex]t_0 = x_1[/itex]) with [itex]x_n = p_{n,n-1}[/itex], then I think [itex]\sum_{n=1}^{+\infty} P^n(0,0) = \sum_{n=0}^{+\infty} t_n[/itex]. But it seems near impossible to prove that 1 minus this expression equals [itex]\frac{6}{\pi^2}[/itex] so I'm probably way off track...
---
Any better suggestions? How would you approach this problem instead?