Editing Taylor's theorem (section)

=== Proof for Taylor's theorem in one real variable ===

Let<ref>{{harvnb|Stromberg|1981}}</ref>

<math display="block"> h_k(x) = \begin{cases}
\frac{f(x) - P(x)}{(x-a)^k} & x\not=a\\
0&x=a
\end{cases}
</math>

where, as in the statement of Taylor's theorem,

<math display="block"> P(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k.</math>

It is sufficient to show that

<math display="block"> \lim_{x\to a} h_k(x) =0. </math>

The proof here is based on repeated application of [[L'Hôpital's rule]].  Note that, for each <math display="inline">j=0,1,...,k-1</math>, <math>f^{(j)}(a)=P^{(j)}(a)</math>.  Hence each of the first <math display="inline">k-1</math> derivatives of the numerator in <math>h_k(x)</math> vanishes at <math>x=a</math>, and the same is true of the denominator.  Also, since the condition that the function <math display="inline">f</math> be <math display="inline">k</math> times differentiable at a point requires differentiability up to order <math display="inline">k-1</math> in a neighborhood of said point (this is true, because differentiability requires a function to be defined in a whole neighborhood of a point), the numerator and its <math display="inline">k-2</math> derivatives are differentiable in a neighborhood of <math display="inline">a</math>. Clearly, the denominator also satisfies said condition, and additionally, doesn't vanish unless <math display="inline">x=a</math>, therefore all conditions necessary for L'Hôpital's rule are fulfilled, and its use is justified. So

<math display="block">\begin{align}
\lim_{x\to a} \frac{f(x) - P(x)}{(x-a)^k}
&= \lim_{x\to a} \frac{\frac{d}{dx}(f(x) - P(x))}{\frac{d}{dx}(x-a)^k} \\[1ex]
&= \cdots \\[1ex]
&= \lim_{x\to a} \frac{\frac{d^{k-1}}{dx^{k-1}}(f(x) - P(x))}{\frac{d^{k-1}}{dx^{k-1}}(x-a)^k}\\[1ex]
&= \frac{1}{k!}\lim_{x\to a} \frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a}\\[1ex]
&=\frac{1}{k!}(f^{(k)}(a) - P^{(k)}(a))
= 0
\end{align}</math>

where the second-to-last equality follows by the definition of the derivative at <math display="inline"> x=a</math>.