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.