In Resnick's Adventures of Stochastic Processes, the author gives in page 411 a sufficient conditions for the birth-death process to be regular (a.s. that there are no infinite transitions in finite time):
$$\infty=\sum_{j\in S}\frac{N^{(j)}}{\lambda_j+\mu_j} $$
where $\mu_j$ is the death rate, $\lambda_j$ the birth rate, and $N^{(j)}=\sum_{n\geq 1}1_{[X_n=j]}$ is the number of visits to state $j$.
Then, he says that a sufficient condition for this is $\sum_{j\in S}\frac{1}{\lambda_j}$. Why is that?
I think he meant $\sum_{j\in S}\frac{1}{\lambda_j+\mu_j}$. But then, why would we have $N^{(j)}\geq 1$ for every state? The discrete markov chain associated to the process is recurrent and irreducible, but it's also with countably infinite states...
Any help would be appreciated.