An homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with discrete state space $\mathcal{S}$. Consider the minimum number of steps to visit $k\in \mathcal{S}$, $$\tau_{k}:=\min \left\{n\ge 1:\, X_n=k \right\}.$$ where $\tau_{k}$ is defined to be $\infty$, when doesn't exist any $n$ such that $X_n=k$, called the first passage time to get to state $k$.
For any $i\in\mathcal{S}$,set the probabilities of visiting $k$ at step $n$ (starting) from $i$ , $$f^{(n)}_{ik}:=\mathbf{P}(\tau_{k}=n\mid X_0=i)=\mathbf{P}(X_n=k,x_{n-1}\ne=k,\cdots,X_1\ne k\mid X_0=i).$$ $$f_{ik}:=\mathbf{P}(\tau_{k}<\infty \mid X_0=i)=\sum_{n=1}^{\infty}\mathbf{P}(\tau_{k}=n\mid X_{0}=i)=\sum_{n=1}^{\infty}f^{(n)}_{ik}.$$
If $X_0=i$, then the mean first passage time to state $k$,denoted $\mu_{ik}$,is defined as $$\mu_{ik}:=\sum_{n=1}^{\infty}nf^{(n)}_{ik}.$$ This definition can be extended to include the case $f_{ik}=\sum_{n=0}^{\infty}f^{n}_{ik}<1$ by defining the probability of never reaching state $k$ from state $i$ as $1-f_{ik}$.
$(1)$ Prove that if $f_{ik}<1$, then the mean first passage time is infinite;
$(2)$ Let $i,k\in\mathcal{S}$, prove that $\mu_{ik}=1+\sum_{j\ne i}p_{ij}\mu_{jk}$.
$(1)$ If $f_{ik}=\sum_{n=0}^{\infty}f^{n}_{ik}=1$, $\mu_{ik}=\mathbf{E}(\tau_{k}\mid X_0=i)$. $f_{ik}<1$ means $\mathbf{P}(\tau_{k}=\infty \mid X_0=i)>0$, then $\mathbf{E}(\tau_{k}\mid X_0=i)=\infty$. But in the case $f_{ik}<1$, $\mu_{ik}\ne\mathbf{E}(\tau_{k}\mid X_0=i)$. How to get $\mu_{jk}$ is infinite?
$(2)$ I think this equality should be $\mu_{ik}=f_{ik}+\sum_{j\ne i}p_{ij}\mu_{jk}$, I don't understand why $\mu_{ik}=1+\sum_{j\ne i}p_{ij}\mu_{jk}$?