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

Thus, defining

(as it always is), we have derived the
following

polynomial approximation:

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.

**Next Section:** Taylor Series with Remainder**Previous Section:** Appendix: Frequencies Representable
by a Geometric Sequence