## Informal Derivation of Taylor Series

We have a function and we want to approximate it using an
th-order *polynomial*:

*remainder term*. We may assume and are

*real*, but the following derivation generalizes unchanged to the complex case.

Our problem is to find fixed constants so as to obtain the best approximation possible. Let's proceed optimistically as though the approximation will be perfect, and assume for all ( ), given the right values of . Then at we must have

where denotes the th derivative of with respect to , evaluated at . Solving the above relations for the desired constants yields

Thus, defining (as it always is), we have derived the following polynomial approximation:

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Taylor Series with Remainder

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Appendix: Frequencies Representable by a Geometric Sequence