Consider a birth-death process on $\mathbb{N}_0$ with transition probabilities given by $$p_{0, 1} = 1, p_{i, i - 1} + p_{i, i + 1} = 1, p_{i, i + 1} = \left(\frac {i + 1} i\right)^2p_{i, i - 1}, i \geq 1.$$ Assuming $X_0 = 0$, calculate the probability of the event $$\{X_n \geq 1\ \forall\ n \geq 1\}.$$
Hint: $$\sum^{\infty}_{m = 1} \frac 1 {m^2} = \frac {\pi^2} 6.$$
My thoughts
Firstly, I know that starting at $X_0 = 0$ is equivalent to starting at $X_1 = 1$. Secondly, it is also trivial to obtain $$p_{i, i - 1} = \frac {i^2} {(i + 1)^2 + i^2}$$ and $$p_{i, i + 1} = 1 - \frac {i^2} {(i + 1)^2 + i^2}.$$
Edit
Following a comment below, since we already know it is irreducible, I think one way to proceed would be to determine whether the chain is transient or recurrent.
Now, define $$\gamma_0 = 1$$ and $$\gamma_y = \frac {\prod\limits^y_{i = 1} p_{i, i - 1}} {\prod\limits^y_{i = 1} p_{i, i + 1}}\ \forall\ y \geq 1.$$ From what I was taught, the chain will be recurrent if and only if $$\sum^{\infty}_{z = 0} \gamma_z = \infty.$$
However, I am not sure how to proceed. Am I even on the right track here?
As my professor has only briefly touched on birth-death processes and this is my first time encountering such a problem, any intuitive suggestions will be greatly appreciated :)