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## 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.

**Previous:** Real Exponents**Next:** Imaginary Exponents

**About the Author: ** Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at

Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See

http://ccrma.stanford.edu/~jos/ for details.