Find the coefficient of $x^3$ in the product of two power series: $(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots)(1 + x + x^2 + x^3 + \cdots)$. (Hint: the first series is $e^x$ and the second is $\frac{1}{1-x}$.)