##
A First Look at Taylor Series

Most ``smooth'' functions

can be expanded in the form of a

*Taylor series expansion*:

This can be written more compactly as

where `

' is pronounced ``

factorial''.
An informal derivation of this formula for

is given in
Appendix

E. 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 below

kHz, and the audible

spectrum of any sum of

sinusoids is infinitely differentiable. (Recall
that

and

,
etc.). See §

E.6 for more about this point.

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