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Series Expansions

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

$\displaystyle f(x) = f(0) + \frac{f^\prime(0)}{1}(x) + \frac{f^{\prime\prime}(0...
...cdot 2}(x)^2 + \frac{f^{\prime\prime\prime}(0)}{1\cdot 2\cdot 3}(x)^3 + \cdots,$ (7.18)

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$.
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