I have a Markov chain with transition probabilities $t_{i,i+1} = \binom{k+i}{k}^{-1}$ and $t_{i,0} = 1-t_{i,i+1}$, i.e. we have an absorbing chain with absorption probability approaching one as $i \rightarrow \infty$. I feel pretty confident you should be able to approximate this with an $N$ state Markov chain, construct its transition matrix $T_N$ and then simply let $N$ go to infinity. For example, the average time $\mathbb{E}[\tau]$ to absorption for the countably infinite state Markov chain would hopefully be okay to write as
$$\mathbb{E}[\tau] = \lim_{N \rightarrow \infty} \boldsymbol{\pi}_0^TT_N \textbf{1}$$
completely analogous to a finite state Markov chain. But I want to be certain all the qualifications for convergence are satisfied. I'm guessing it's not enough for the transient probabilities to go to zero, they need to do so at a brisk enough pace. So I'm wondering what the requirements are?