Given transition probability as below, how can I tell that the Markov Chain $X_n$ is a positive recurrent, null recurrent or transient?
$$p(x,0)=1/(x^2+1),\quad p(x,x+1)=(x^2+1)/(x^2+2)$$ state space $S=\{0,1,2,...\}$
I have tried to calculate $\alpha(x)$ for $x \in S$, but it turns out to be $1$ for all $x$. So it is recurrent. Then try to find invariant probability distribution, but meet some trouble here.