Problem:
Let $p_{0}(x), p_{1}(x), p_{2}(x), \ldots$ be polynomials such that $p_{0}(x) = x$ and for all positive integers $n$, $\frac{d}{dx} p_{n}(x) = p_{n-1}(x)$. Define the function $p(x): [0, \infty) \to \mathbb{R}$ by $p(x) = p_{n}(x)$ for all $x \in [n, n+1]$. Given that $p(x)$ is continuous on $[0, \infty)$, compute
$$\sum_{n=0}^{\infty} p_{n}(2009)$$