A First Look at Taylor Series

Most ``smooth'' functions $f(x)$ can be expanded in the form of a Taylor series expansion:

$$f(x) = f(x_0) + \frac{f^\prime(x_0)}{1}(x-x_0) + \frac{f^{\prime... ...^2 + \frac{f^{\prime\prime\prime}(x_0)}{1\cdot 2\cdot 3}(x-x_0)^3 + \cdots .$$

This can be written more compactly as

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n,$$

where $n!$ is pronounced ``$n$ factorial''. An informal derivation of this formula for $x_0=0$ is given in Appendix A. Clearly, since many derivatives are involved, a Taylor series expansion is only possible when the function is so smooth that it can be differentiated again and again. Fortunately for us, all audio signals are in that category, because hearing is bandlimited to $20$ kHz, and the audible spectrum of any sum of sinusoids is infinitely differentiable. (Recall that $\sin^\prime(x)=\cos(x)$ and $\cos^\prime(x)=-\sin(x)$, etc.). See §A.6 for more about this point.