Let $P_1(x)=\frac{1}{x}$ and $P_n(x)=P_{n-1}(x)+P_{n-1}(x-1)$ for every natural $ n$ greater than $1$. Find the value of $P_{2008}(2008)$. (Mathophile)
Problem
Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p5 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
Tags: algebra, polynomial, recurrence relation