I am looking at the generating functions and I have stumbled upon the following result:
$$Q(S)=\frac{1-P(s)}{1-s}$$
where $P(s)=p_0+p_1s+p_2s^2+...$ and $Q(s)=q_0+q_1s+q_2s^2+..$ and $P\{X=j\}=p_j$ and $P\{X>j\}=q_j$, where $q_k=p_{k+1}+p_{k+2}+...$
I can follow all of the above, then the book follows to state: "Applying the mean value theorem to the numerator",(the equation for Q(s))"we see that $Q(s)=P'(\sigma)$. I do not know what they mean by the applying the mean value theorem part at all, and how they get the result. Could you please clarify?
Also: $P'(s) = \sum_i^{\infty}kp_ks^{k-1}$