Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$. Find $ \lim_{n\to\infty}(a_0+a_1+\ldots+a_n)$.
Problem
Source: Moldova National MO 2008, 12 Grade, Problem 4
Tags: limit, real analysis, real analysis unsolved