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

where
, the approximation error, is called the
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

That's one constant down and to go! Now let's look at the first
derivative of with respect to , again assuming that
:

Evaluating this at gives

In the same way, we find

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

This is the th-order Taylor series expansion of about the
point . Its derivation was quite simple. The hard part is
showing that the approximation error (remainder term
) is
small over a wide interval of values. Another ``math job'' is to
determine the conditions under which the approximation error
approaches zero for all as the order goes to infinity. The
main point to note here is that the Taylor series itself is simple to
derive.