Taylor expansion/R/One variable/Introduction/Section

So far, we have only considered power series of the form ${\displaystyle {}\sum _{k=0}^{\infty }c_{k}x^{k}}$. Now we allow that the variable ${\displaystyle {}x}$ may be replaced by a "shifted variable“ ${\displaystyle {}x-a}$, in order to study the local behavior in the expansion point ${\displaystyle {}a}$. Convergence means, in this case, that some ${\displaystyle {}\epsilon >0}$ exists, such that for

${\displaystyle {}x\in ]a-\epsilon ,a+\epsilon [\,,}$

the series converges. In this situation, the function, presented by the power series, is again differentiable, and its derivative is given as in fact. For a convergent power series

${\displaystyle {}f(x):=\sum _{k=0}^{\infty }c_{k}(x-a)^{k}\,,}$

the polynomials ${\displaystyle {}\sum _{k=0}^{n}c_{k}(x-a)^{k}}$ yield polynomial approximations for the function ${\displaystyle {}f}$ in the point ${\displaystyle {}a}$. Moreover, the function ${\displaystyle {}f}$ is arbitrarily often differentiable in ${\displaystyle {}a}$, and the higher derivatives in the point ${\displaystyle {}a}$ can be read of from the power series directly, namely

${\displaystyle {}f^{(n)}(a)=n!c_{n}\,.}$

We consider now the question whether we can find, starting with a differentiable function of sufficiently high order, approximating polynomials (or a power series). This is the content of the Taylor expansion.

Definition

Let ${\displaystyle {}I\subseteq \mathbb {R} }$ denote an interval,

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

an ${\displaystyle {}n}$-times differentiable function, and ${\displaystyle {}a\in I}$. Then

${\displaystyle {}T_{a,n}(f)(x):=\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}\,}$
is called the Taylor polynomial of degree ${\displaystyle {}n}$ for ${\displaystyle {}f}$ in the point ${\displaystyle {}a}$.

So

${\displaystyle {}T_{a,0}(f)(x):=f(a)\,}$

is the constant approximation,

${\displaystyle {}T_{a,1}(f)(x):=f(a)+f'(a)(x-a)\,}$

is the linear approximation,

${\displaystyle {}T_{a,2}(f)(x):=f(a)+f'(a)(x-a)+{\frac {f^{\prime \prime }(a)}{2}}(x-a)^{2}\,}$

${\displaystyle {}T_{a,3}(f)(x):=f(a)+f'(a)(x-a)+{\frac {f^{\prime \prime }(a)}{2}}(x-a)^{2}+{\frac {f^{\prime \prime \prime }(a)}{6}}(x-a)^{3}\,}$

is the approximation of degree ${\displaystyle {}3}$, etc. The Taylor polynomial of degree ${\displaystyle {}n}$ is the (uniquely determined) polynomial of degree ${\displaystyle {}\leq n}$ with the property that its derivatives and the derivatives of ${\displaystyle {}f}$ at ${\displaystyle {}a}$ coincide up to order ${\displaystyle {}n}$.

Theorem

Let ${\displaystyle {}I}$ denote a real interval,

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

an ${\displaystyle {}(n+1)}$-times differentiable function, and ${\displaystyle {}a\in I}$ an inner point of the interval. Then for every point ${\displaystyle {}x\in I}$, there exists some ${\displaystyle {}c\in I}$ such that

${\displaystyle f(x)=\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}+{\frac {f^{(n+1)}(c)}{(n+1)!}}(x-a)^{n+1}.}$
Here, ${\displaystyle {}c}$ may be chosen between

${\displaystyle {}a}$ and ${\displaystyle {}x}$.

Proof

This proof was not presented in the lecture.
${\displaystyle \Box }$

Corollary

Suppose that ${\displaystyle {}I}$ is a bounded closed interval,

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

is an ${\displaystyle {}(n+1)}$-times continuously differentiable function, ${\displaystyle {}a\in I}$ an inner point, and ${\displaystyle {}B:={\max {\left(\vert {f^{(n+1)}(c)}\vert ,c\in I\right)}}}$. Then, between ${\displaystyle {}f(x)}$ and the ${\displaystyle {}n}$-th Taylor polynomial, we have the estimate

${\displaystyle {}\vert {f(x)-\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}}\vert \leq {\frac {B}{(n+1)!}}\vert {x-a}\vert ^{n+1}\,.}$

Proof

The number ${\displaystyle {}B}$ exists due to fact, since the ${\displaystyle {}(n+1)}$-th derivative ${\displaystyle {}f^{(n+1)}}$ is continuous on the compact interval ${\displaystyle {}I}$. The statement follows, therefore, directly from fact.

${\displaystyle \Box }$