Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 8 / a / ii / Solution

By hypothesis the power series converges at every positive real argument, so its radius of convergence is infinite. Termwise differentiation of a power series is therefore valid in a neighborhood of $s=1$. For $k\ge 1$,At $s=1$, the right side is the expectation of the falling factorial of $X$:The special case $k=1$ givesFor $k=0$, take the empty product as $1$; the equality becomes $G(1)=1$. These are the factorial moments.