$$ f_{i j}^{n}=P\left\{X_{n}=j, X_{v} \neq j, v=1,2, \ldots, n-1 \mid X_{0}=i\right\} $$ That is the probability of the first passage from state i, to state j at the nth transition(Note:$_{ij}^{0}$ is defined as 0, so start from 0 or 1 is the same thing)
Now we are given $$ P_{i j}^{n}=\sum_{k=0}^{n} f_{i j}^{k} P_{j j}^{n-k}, \quad i \neq j, n \geq 0 $$ $$ P_{i i}^{n}=\sum_{k=0}^{n} f_{i i}^{k} P_{i i}^{n-k} . \quad n \geq 1 $$
If $ \sum_{n=1}^{\infty} f_{jj}^{n}<1 $, how can we find the limit $\lim_{n\to \infty}P_{ij}^{n}$