Informal Derivation of Taylor Series
We have a function and we want to approximate it using an
th-order polynomial:




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