Understanding equations representing time spent in certain Markov states My notes on Markov chain long-term behaviour present the following definition:

Definition 10.
For any state $j$, define $N_j$ to be the number of times the Markov chain is at state $j$, excluding the initial state. Formally, $N_j = \# \{ n \ge 1 : X_n = j \}$.
Note that $N_j$ takes the values in $\mathbb{N} \cup \{ 0 \} \cup \{ \infty \}$.
Note also that
$$P(N_j = m \vert X_0 = i) = f_{ij} f_{jj}^{m - 1}(1 - f_{jj})$$
and
$$P(N_j \ge m \vert X_0 = i) = f_{ij} f_{jj}^{m - 1}.$$
It follows that
$$P(N_j = \infty \vert X_0 = i) = \begin{cases} 0 & \text{if} \ f_{jj} < 1, \\ f_{ij} & \text{if} \ f_{jj} = 1. \end{cases}$$
So $P(N_j = \infty \vert X_0 = i) = 0$ if $j$ is transient, and $f_{ij}$ if $j$ is recurrent.

I have spent a significant amount of time reading over these notes (see below) to try and gain a good mathematical and intuitive understanding for the equations $P(N_j = m \vert X_0 = i) = f_{ij} f_{jj}^{m - 1}(1 - f_{jj})$, $P(N_j \ge m \vert X_0 = i) = f_{ij} f_{jj}^{m - 1}$, and $P(N_j = \infty \vert X_0 = i) = \begin{cases} 0 & \text{if} \ f_{jj} < 1, \\ f_{ij} & \text{if} \ f_{jj} = 1 \end{cases}$, but I am having great difficulty doing so. The way these concepts are explained in my notes make it very difficult for me to develop a clear understanding of this. For the first two equations in particular, I want to understand the presence of the terms in $f_{ij} f_{jj}^{m - 1}(1 - f_{jj})$ and $f_{ij} f_{jj}^{m - 1}$, as explained/introduced in the notes (see below).
The full notes are as follows:

Definition 7.
For a non-empty subset $A$ of the state space, define
$$T_A = \min\{ n \ge 1 : X_n \in A \}.$$
$T_A$ is called the hitting time of the time of first passage.
If $X_n \not\in A$ for all $n \ge 1$, then $T_A = \infty$.
For a state $i$, $T_{\{i\}}$ is written as $T_i$.
Theorem 10.
$$p^{(n)}_{ij} = \sum_{m = 1}^n P(T_j = m \vert X_0 = i) p^{(n - m)}_{jj}.$$
Definition 8.
For states $i$ and $j$, define
$$f^{(n)}_{ij} = P(X_n = j, X_k \not= j, k = 1, 2, \dots, n - 1 \vert X_0 = i)$$
and $f_{ij} = \sum_{n = 1}^\infty f^{(n)}_{ij}$.
In other words, $f^{(n)}_{ij}$ is the probability that, starting from state $i$, the first visit to $j$ occurs at the $n$th transition.
$f^{(1)}_{ij} = p_{ij}$. And by convention, $f^{(0)}_{ij} = 0$, including for the case $i = j$.
$f_{ij}$ is the same as $P(T_j < \infty \vert X_0 = i)$, and is the probability of the Markov chain reaching state $j$ some time if it starts from state $i$.
Note that $P(T_j = \infty \vert X_0 = i)$ is $1 - f_{ij}$. In addition, $i$ leads to $j$ if and only if $f_{ij} > 0$.
Theorem 11.
$$p^{(n)}_{ii} = \sum_{k = 0}^n f_{ii}^{(k)}p^{(n - k)}_{ii}$$
Definition 9. A state $i$ is said to be recurrent if $f_{ii} = 1$, and transient if $f_{ii} < 1$.
If the state $i$ is recurrent, the probability that the Markov chain starting from $i$ returns to $i$ is $1$.
If $i$ is transient, there is a positive probability that the chain, starting from $i$, will never return to $i$.
All absorbing states are recurrent, because if $p_{ii} = 1$, then $f_{ii} = 1$.
Definition 10.
For any state $j$, define $N_j$ to be the number of times the Markov chain is at state $j$, excluding the initial state. Formally, $N_j = \# \{ n \ge 1 : X_n = j \}$.
Note that $N_j$ takes the values in $\mathbb{N} \cup \{ 0 \} \cup \{ \infty \}$.
Note also that
$$P(N_j = m \vert X_0 = i) = f_{ij} f_{jj}^{m - 1}(1 - f_{jj})$$
and
$$P(N_j \ge m \vert X_0 = i) = f_{ij} f_{jj}^{m - 1}.$$
It follows that
$$P(N_j = \infty \vert X_0 = i) = \begin{cases} 0 & \text{if} \ f_{jj} < 1, \\ f_{ij} & \text{if} \ f_{jj} = 1. \end{cases}$$
So $P(N_j = \infty \vert X_0 = i) = 0$ if $j$ is transient, and $f_{ij}$ if $j$ is recurrent.

I was wondering if people would please take the time to explain these three equations, both mathematically and intuitively, so that the ideas conveyed by them are clear to me.
 A: *

*We begins from state $i$ and we want to visit state $j$ exactly $m$ times. 


*

*Begining from state $i$, I have to be able to visit state $j$, that happens with probability $f_{ij}$, when that happens, that is the first visit. Probability in this step is $f_{ij}$

*Now starting from state $j$, we want to visit state $j$ for another $m-1$ times, the probability of each such occurence is $f_{jj}$, it has to happen $m-1$ times. Coresponding probability: $f_{jj}^{m-1}$

*After that, starting from state $j$, we want to compute the probability that it never visit state $j$ again. Corresponding probability: $1-f_{jj}$.



Multiplying them together
\begin{align}
P(N_j=m|X_0=i) = f_{ij}f_{jj}^{m-1}(1-f_{jj})
\end{align}



*

*We begins from state $i$ and we want to visit state $j$ at least $m$ times.


The argument is the same as earlier, that is 


*

*Begining from state $i$, I have to be able to visit state $j$, that happens with probability $f_{ij}$, when that happens, that is the first visit. Probability in this step is $f_{ij}$

*Now starting from state $j$, we want to visit state $j$ for another $m-1$ times, the probability of each such occurence is $f_{jj}$, it has to happen $m-1$ times. Coresponding probability: $f_{jj}^{m-1}$

*What happens next is irrelevant. We have achieved our goal of visiting $j$ at least $m$ times.


\begin{align}
P(N_j\ge m|X_0=i) = f_{ij}f_{jj}^{m-1}
\end{align}



*

*Given that $f_{jj}<1$, we want to show that $P(N_j = \infty|X_0=i)=0$.


We begins from state $i$, first we need to reach state $j$, and then we will visit state $j$ infinitely many times. However, if we are told that $f_{jj}<1$, then we know that $j$ is a transient state, hence after certain time, we will never visit state $j$ again, hence we can't have $N_j=\infty$. 
A more mathematical claim is by taking the limit of the previous term:
$$\lim_{m \to \infty}P(N_j\ge m|X_0=i)=\lim_{m \to \infty}f_{ij}f_{jj}^{m-1}=f_{ij}\lim_{m \to \infty}f_{jj}^{m-1}=0$$
by geometric series.



*

*Given $f_{jj}=1$, we want to show that $P(N_j = \infty|X_0=i)=0$
We begin from state $i$, we need to visit state $j$, once we visit it, since state $j$ is a recurrence state, it will be visited infinitly often.
$$\lim_{m \to \infty}P(N_j\ge m|X_0=i)=\lim_{m \to \infty}f_{ij}f_{jj}^{m-1}=f_{ij}\lim_{m \to \infty}1^{m-1}=f_{ij}$$
