It's about the definition of a recurrent state. I think there might be something wrong in this book. it is said as following.
Suppose a Markov chain starts in state i. State i is said to be a recurrent state if the Markov chain returns to state i with probability 1; that is,
pi =P(ever returning to state i) =1
If the probability pi is less than 1, state iis said to be a transientstate (Leon-Garcia, 1994). If the Markov chain starts in a recurrent state, that state reoccurs an infinite number of times. If it starts in a transient state, that state reoccurs only a finite number of times, which may be explained as follows: We may view the reoccurrence of state i as a Bernoulli trial with a probability of success equal to pi .The number of returns is thus a geometric random variable with a mean of (1 - 1/pi). If pi < 1, it follows that the number of an infinite number of successes is zero.Therefore, a transient state does not reoccur after some finite number of returns.
You can see those italics. I can't figure out what kind of the Bernoulli trail it is. Why it has a mean of (1 - 1/pi), which indicates that when pi < 1, the mean is less than zero. Maybe it is a typo, maybe it should have meant to be (1/pi - 1). But it still seems make no sense. I mean if you want to prove that the probability for a transient state to reoccur an infinite number of times is zero, you can just use the limitation of pi^n is zero when pi<1. It has really confused me for quite a long time. Anyone has any advice here?