For example, from calculus we know that the power series \(1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots + \frac{x^n}{n!} + \cdots\) converges to the function \(e^x\text{.}\) So we can use \(e^x\) as a way of talking about the sequence of coefficients of the power series for \(e^x\text{.}\) When we write down a nice compact function which has an infinite power series that we view as a generating series, then we call that function a generating function. In this example, we would say