Series Expansions

Any ``smooth'' function $f(x)$ can be expanded as a Taylor series expansion:

$$f(x) = f(0) + \frac{f^\prime(0)}{1}(x) + \frac{f^{\prime\prime}(0... ...cdot 2}(x)^2 + \frac{f^{\prime\prime\prime}(0)}{1\cdot 2\cdot 3}(x)^3 + \cdots,$$ (S.2)

where ``smooth'' means that derivatives of all orders must exist over the range of validity. Derivatives of all orders are obviously needed at $x=0$ by the above expansion, and for the expansion to be valid everywhere, the function $f(x)$ must be smooth for all $x$.