If $Z$ is a Markov chain with transition matrix $P$ where there exist some $k$ of moving from a recurrent state $j$, to a transient state $i$, by $f_{ij}^{(k)} >0$. Use $$p^{(n)}_{(i,j)}=\sum^n_{m=0}f^{(m)}_{i,j}p^{(n-m)}_{j,j}$$ to show that $$\sum^\infty_{n=1} p^{(n)}_{i,j} = \infty$$
My idea is that since $f_{ij}^{(k)} >0$ and $j$ is recurrent, then $i$ is also recurrent and they $i$ communicates with $j$. Thus,
$$\sum^\infty_{n=1} p^{(n)}_{i,j} = \sum^\infty_{n=1} p^{(n)}_{j,j}= \infty$$ But I am not sure about how I did it. Help needed, thanks!