Section 2.   Definitions

Definition: Given a function f(x) whose first n derivatives can be found, we define the Taylor polynomial P_n(x) of degree n for the function f(x) about the value c as

where denotes the n^th derivative of the function f evaluated at x=c .

The Taylor series associated with f about c is simply the Taylor polynomial P_n(x) for f about c where  . More precisely, we have

This is an impressive equality, to say the least. However, in practice, we will not deal with power series as often as with a polynomial P_n(x) , which is simply a truncated version of the power series. In that case, we will see that