Theorem: i is recurrent then $\mu_{ii}=\sum_{n=1}^\infty nf_{ii}^n = \lim_{s\rightarrow 1^{-}} \frac 1 {(1-s)p_{ii}(s)}$
Proof:
Let $i$ be recurrent.
Then, note:
\begin{align} & \lim_{s\rightarrow 1^{-}} F_{ii} (s))=\lim_{s\rightarrow 1^{-}} \sum_{n=0}^\infty f_{ii}^n s^n \\[10pt] & \Longrightarrow F_{ii}(\approx 1^{-})=\sum_{n=0}^\infty f_{ii}^n=1 \\[10pt] & \Longrightarrow \lim_{s\rightarrow 1^{-}} ( 1-F_{ii} (s)) =0 \end{align}
Again, observe:
\begin{align} & F_{ii} (s)=\sum_{n=0}^\infty f_{ii}^n s^n \\[10pt] \Longrightarrow & F_{ii}'(s)=\sum_{n=0}^\infty nf_{ii}^n s^{n-1} \end{align}
so,
$$F_{ii}'(\approx 1^{-})=\sum_{n=0}^\infty nf_{ii}^n \approx \lim_{s\rightarrow 1^{-}} F_{ii}' (s)=\lim_{s\rightarrow 1^{-}} \sum_{n=0}^\infty nf_{ii}^n s^{n-1} $$
How to I then conclude that $F_{ii}'(s)=\lim_{s\rightarrow 1^{-}} \frac{1-F_{ii} (s)}{1-s}?$
Vital hints are appreciated.