Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 43 / 1 / c / Solution

Here $a+b/n=a(n+m-1)/n$, so iteration givesThe negative-binomial series ${\sum }_{n\ge 0}\left(\genfrac{}{}{0ex}{}{m+n-1}{n}\right)a^n=(1-a{)}^{-m}$, valid for $0<a<1$, normalizes this toThis is the negative binomial distribution counting failures before the $m$th success, with success probability $1-a$. The convention is specified because a negative-binomial variable is sometimes defined to count trials instead. Its count expected value is $ma/(1-a)$, and the Panjer recursion begins at $g_0=(1-a{)}^m$. For $m=1$ this reduces to the failure-count geometric distribution.