A First Look at Taylor Series

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). A First Look at Taylor Series

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A First Look at Taylor Series

Any “smooth” function $f(x)$ can be expanded in the form of a Taylor series:

\

This can be written more compactly as
\

An informal derivation of this formula for $x_0=0$ is given in §3.2 and §3.3. 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 can be defined so as to be in that category. This is because hearing is bandlimited to$20$ kHz, and any sum of sinusoids up to some maximum frequency, i.e., any audible signal, is infinitely differentiable. (Recall that $\ and $\, etc.). See §3.6 for more on this topic.

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