Markov Chains state classification

Just learning about Markov Chains and the classification of states. We were told [tex] \displaystyle\sum_{n=1}^\infty p_{ij} (n)=\infty [/tex] if the state is recurrent and [tex]/f _{ij} \g 0 [/tex].

But i understand [tex]/p_{ij} (n)[/tex] to be the probability of getting to state j from i at stage n. So the sum of these probabilities is infinite? I'm used to probabilities summing to 1. It would appear that this infinite sum is not a probability in same sense i am used to. Are we just summing - how shall i say? - disconnected - probabilties?

Also, in what sense can we say that a null recurrent state is different to a transient state, if both have an infinite mean recurrence state? Just seems like different ways of getting the same result.