Why is the probability of ever making a transition to state $j$ starting from $i$ equals to $\sum_{n=1}^{\infty}f_{ij}^n$? This trouble comes from:

I did not understand why this probability would not be:
$f_{ij}=\displaystyle\sum_{n=1}^{\infty}P_{ij}^{n}$
Where:
$P^{n}_{ij}=P(X_n=j|X_0=i)$
I see no problem using $P^{n}_{ij}$ instead the math seems would hold.

$P_{ij}^n$ is the probability that after $n$ stages we are in state $j$ given that we started in state $i$, whereas $f_{ij}^n$ is the probability that the first transition from state $i$ to state $j$ occurs in exactly $n$ stages.  The first is key.
Consider a 2-state Markov chain where the probability of transitioning from state $0$ to state $1$ at stage $1$ is $0.1$, and after stage $1$ we remain in whichever state we are in forever (i.e., $P$ is a $2 X 2$ identity matrix).  $P_{0,1}^n = 0.1$ for all $n$, which obviously gives $f_{0,1} = \infty$ if we use
$f_{ij}=\displaystyle\sum_{n=1}^{\infty}P_{ij}^{n}$.  Hmmm....
On the other hand, the probability that the first transition from $0$ to $1$ occurs at stage $n > 1$ equals 0, since the only way you can get to state $1$ for $n>1$ is to have been there at stage $n-1$, so it can't be the first transition to state $1$.  The probability that the first transition to state $1$ occurs at $n=1$ is $0.1$, as defined above, and summing the first transition probabilities obviously gives the correct answer of $0.1$.
For a more intuitive response - summing the $P_{ij}^n$ gives you a sum of the probabilities that you are in state $j$ at stage $n$ given you started at state $i$, which equals the expected number of stages during which you are in state $j$ given that you started in state $i$. This isn't what you want (well in this case.)  However, in order to ever be in state $j$ given you have have started in state $i$, there has to be a first time you are in state $j$.  The probability that you ever make the transition to state $j$ $= 1 - $ the probability that you never make the transition to state $j$, in other words, that there never is a first time you are in state $j$,  which latter probability is just $1 - f_{ij}^1 - f_{ij}^2 - \dots$, i.e., $1 - $ the sum of the probabilities that each individual stage is the first passage stage.  And... $1 - (1 - \Sigma_n f_{ij}^n) = \Sigma_n f_{ij}^n$, the probability that you ever transition to state $j$.
