Motivating Taylor Polynomials Via the Binomial Theorem
Dobbs, David E., Mathematics and Computer Education
(ProQuest: ... denotes formulae omitted.)
The material of this note can find classroom use when students are studying Taylor polynomials in a Calculus or Elementary Analysis course. Taylor polynomials play an important role in calculus and its applications. For instance, it was shown in [3, Theorem 2] that Taylor polynomials can arise naturally in a differential equations course when one seeks polynomial approximations to power series solutions of certain initial value problems by using the method of undetermined coefficients. That paper was motivated by an example in a calculus reform textbook [1, Example 14, pp. 587-588]. One calculus textbook that is widely used today motivates the introduction of the nth Taylor polynomial as the polynomial of degree at most ? that gives the "best approximation" (in a certain sense) to a given infinitely differentiable function [4, p. 254] and, later, as a partial sum of the Taylor series of the given function [4, p. 607] . The main purpose of the present note is to motivate the supplement of the study of Taylor polynomials by means of the Binomial Theorem. We next explain how doing so in the typical calculus course would have at least three advantages over the other strategies that were described above.
First, our proposed method uses a tool that is already familiar, as the Binomial Theorem is typically used early in a calculus course to prove the formula for the derivative of x^sup n^, as in [4, p. 184]. Second, our method reinforces the standard rules for differentiating sums or constant multiples of differentiable functions, which are generally covered shortly after the derivative of x^sup n^, as in [4, pp. 186-187]. Third, our method does not need time-consuming calculations of n01 derivatives. Thus, in regard to the third point, this note contributes to the view expressed in  that the Taylor polynomials/series of the most useful functions can often be obtained without the explicit calculations of higher derivatives of the kind that are found in [4, pp. 610-612].
2. THE BINOMIAL THEOREM IMPUES TAYLOR'S THEOREM FOR POLYNOMIALS
We will show how the Binomial Theorem, ... leads to a proof that any nth degree polynomial function, f(x)-a^sub n^x^sup n^+a^sub n-1^x^sup n-1^+ ... + a^sub 0^, can be expressed as ..., which happens to be (the definition of) the ? * Taylor polynomial of f (at x^sub 0^). The proof given below is thus, in effect, a discovery activity that can lead to the definition of the Taylor polynomials/series for any sufficiently/infinitely differentiable function.
Let us begin with the special case f(x) = x^sup n^, for some positive integer n. As in [2, Example 1 (b)], the key is to rewrite x as the sum x^sub 0^ +(x-x^sub 0^) . Then, raising to the n* power and applying the Binomial Theorem, we have
Next, observe that the factor n(n - 1)(n - 2) ... (n - i + 1)(x^sub 0^)^sup n-i^ is the same as f^sup (i)^(x^sub 0^) for me particular function f(x) = x^sup n^. …