Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 17/latex
\setcounter{section}{17}
\zwischenueberschrift{The Taylor formula}
So far, we have only considered power series of the form \mathl{\sum_{k=0}^\infty c_k x^k}{.} Now we allow that the variable $x$ may be replaced by a \anfuehrung{shifted variable}{} $x-a$, in order to study the local behavior in the \stichwort {expansion point} {} $a$. Convergence means, in this case, that some
\mavergleichskette
{\vergleichskette
{ \epsilon
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
exists, such that for
\mavergleichskettedisp
{\vergleichskette
{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
Theorem 16.1
.
For a convergent power series
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ \defeq} { \sum _{ k= 0}^\infty c_k (x-a)^{ k }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
the polynomials \mathl{\sum_{k=0}^n c_k (x-a)^k}{} yield polynomial approximations for the function $f$ in the point $a$. Moreover, the function $f$ is arbitrarily often differentiable in $a$, and the higher derivatives in the point $a$ can be read of from the power series directly, namely
\mavergleichskettedisp
{\vergleichskette
{ 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
\zusatzklammer {or a power series} {} {.}
This is the content of the \stichwort {Taylor expansion} {.}
\inputdefinition
{ }
{
Let
\mavergleichskette
{\vergleichskette
{I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an
interval,
\mathdisp {f \colon I \longrightarrow \R} { }
an $n$-times
differentiable
function,
and
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then
\mavergleichskettedisp
{\vergleichskette
{ T_{ a,n } ( f) ( x )
}
{ \defeq} { \sum_{ k = 0}^{ n } \frac{ f^{( k )}(a)}{ k !} (x-a)^{ k }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
}
So
\mavergleichskettedisp
{\vergleichskette
{ T_{ a,0 } ( f) ( x )
}
{ \defeq} { f(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is the constant approximation,
\mavergleichskettedisp
{\vergleichskette
{ T_{ a,1 } ( f) ( x )
}
{ \defeq} { f(a) + f'(a)(x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is the linear approximation,
\mavergleichskettedisp
{\vergleichskette
{ T_{ a,2 } ( f) ( x )
}
{ \defeq} { f(a) + f'(a)(x-a) + { \frac{ f^{\prime \prime} (a) }{ 2 } } (x-a)^2
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is the quadratic approximation,
\mavergleichskettedisp
{\vergleichskette
{ T_{ a,3 } ( f) ( x )
}
{ \defeq} { 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 $3$, etc. The Taylor polynomial of degree $n$ is the
\zusatzklammer {uniquely determined} {} {}
polynomial of degree $\leq n$ with the property that its derivatives and the derivatives of $f$ at $a$ coincide up to order $n$.
\inputfaktbeweisnichtvorgefuehrt
{Real function/Taylor formula/(n+1)-times continuously differentiable/Lagrange/Fact}
{Theorem}
{}
{
\faktsituation {Let $I$ denote a
real interval,
\mathdisp {f \colon I \longrightarrow \R} { }
an \mathl{(n+1)}{-}times
differentiable
function,
and
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
an inner point of the interval.}
\faktfolgerung {Then for every point
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
there exists some
\mavergleichskette
{\vergleichskette
{c
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mathdisp {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 }} { . }
}
\faktzusatz {Here, $c$ may be chosen between
\mathkor {} {a} {and} {x} {.}}
}
{Real function/Taylor formula/(n+1)-times continuously differentiable/Lagrange/Fact/Proof
\inputfaktbeweis
{Real function/Taylor formula/(n+1)-times continuously differentiable/Estimate for error/Fact}
{Corollary}
{}
{
\faktsituation {Suppose that $I$ is a
bounded
closed interval,
\mathdisp {f \colon I \longrightarrow \R} { }
is an \mathl{(n+1)}{-}times
continuously differentiable
function,
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
an
inner point,
and
\mavergleichskette
{\vergleichskette
{B
}
{ \defeq }{ {\max { \left( \betrag { f^{(n+1)}(c) } , c \in I \right) } }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktfolgerung {Then, between \mathl{f(x)}{} and the $n$-th
Taylor polynomial,
we have the estimate
\mavergleichskettedisp
{\vergleichskette
{ \betrag { f(x) - \sum_{ k = 0}^{ n } \frac{ f^{( k )}(a)}{ k !} (x-a)^{ k } }
}
{ \leq} { \frac{B}{ (n+1)!}\betrag { x-a }^{n+1}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\faktzusatz {}
}
{
The number $B$ exists due to Theorem 11.13 , since the \mathl{(n+1)}{-}th derivative \mathl{f^{(n+1)}}{} is continuous on the compact interval $I$. The statement follows, therefore, directly from Theorem 17.2 .
\zwischenueberschrift{Criteria for extrema}
We have seen in the 15th lecture that it is a necessary condition for a differentiable function to have a local extremum at a point, that its derivative equals $0$ at this point. We give now an important sufficient criterion, which relies on higher derivatives.
\inputfaktbeweis
{Real function/Extrema/Higher derivatives/Fact}
{Theorem}
{}
{
\faktsituation {Let $I$ denote a
real interval,
\mathdisp {f \colon I \longrightarrow \R} { }
an \mathl{(n+1)}{-}times
continuously differentiable
function,
and
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
an inner point of the interval.}
\faktvoraussetzung {Suppose that
\mathdisp {f'(a)= f^{\prime \prime}(a) = \ldots = f^{(n)}(a)=0 \text{ and } f^{(n+1)}(a) \neq 0} { }
is fulfilled.}
\faktuebergang {Then the following statements hold.}
\faktfolgerung {\aufzaehlungdrei {If $n$ is even, then $f$ does not have a
local extremum
in $a$.
} {Suppose that $n$ is odd. In case
\mavergleichskette
{\vergleichskette
{ f^{(n+1)}(a)
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the function $f$ has an
isolated local minimum
in $a$.
} {Suppose that $n$ is odd. In case
\mavergleichskette
{\vergleichskette
{ f^{(n+1)}(a)
}
{ < }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the function $f$ has an
isolated local maximum
in $a$.
}}
\faktzusatz {}
}
{
Under the given conditions, the
Taylor formula
becomes
\mavergleichskettedisp
{\vergleichskette
{ f(x)-f(a)
}
{ =} { \frac{ f^{ (n +1) } ( c )}{ (n +1)! } (x-a)^{ n +1 }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
with some $c$
\zusatzklammer {depending on $x$} {} {}
between
\mathkor {} {a} {and} {x} {.}
Depending on whether
\mathkor {} {f^{(n+1)}(a)>0} {or} {f^{(n+1)}(a) < 0} {}
holds, we have
\zusatzklammer {due to the continuity of the $(n+1)$-th derivative} {} {}
\mathkor {} {f^{(n+1)}(x)>0} {or} {f^{(n+1)}(x) < 0} {}
for
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{ [a-\epsilon,a+\epsilon]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
for a suitable
\mavergleichskette
{\vergleichskette
{\epsilon
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For these $x$, we have
\mavergleichskette
{\vergleichskette
{c
}
{ \in }{ [a-\epsilon,a+\epsilon]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
so that the sign of \mathl{f^{(n+1)}(c)}{} depends on the sign of \mathl{f^{(n+1)}(a)}{.}
For $n$ even, \mathl{n+1}{} is odd and therefore the sign of \mathl{(x-a)^{n+1}}{} changes at
\mavergleichskette
{\vergleichskette
{x
}
{ = }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
\zusatzklammer {for
\mavergleichskette
{\vergleichskette
{x
}
{ < }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the sign is negative, and for
\mavergleichskette
{\vergleichskette
{x
}
{ > }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the sign is positive} {} {.}
Since the sign of \mathl{f^{(n+1)}(c)}{} does not change, the sign of \mathl{f(x)-f(a)}{} is changing. This means that there can not be an extremum.
Suppose now that $n$ is odd. Then \mathl{n+1}{} is even, hence
\mavergleichskette
{\vergleichskette
{ (x-a)^{n+1}
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mavergleichskette
{\vergleichskette
{x
}
{ \neq }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in the neighborhood. This means, in the neighborhood, in case
\mavergleichskette
{\vergleichskette
{ f^{(n+1)}(a)
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
that
\mavergleichskette
{\vergleichskette
{ f(x)
}
{ > }{ f(a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds, and we have an
isolated minimum
in $a$. If
\mavergleichskette
{\vergleichskette
{ f^{(n+1)}(a)
}
{ < }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
then
\mavergleichskette
{\vergleichskette
{ f(x)
}
{ < }{ f(a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds, and we have an
isolated maximum
in $a$.
A special case of this is that in case
\mavergleichskette
{\vergleichskette
{ f'(a)
}
{ = }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ f^{\prime \prime}(a)
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
then we have an isolated minimum, and in case
\mavergleichskette
{\vergleichskette
{ f'(a)
}
{ = }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ f^{\prime \prime}(a)
}
{ < }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have an isolated maximum.
\zwischenueberschrift{The Taylor series}
\inputdefinition
{ }
{
Let
\mavergleichskette
{\vergleichskette
{I
}
{ \subseteq }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an
interval,
\mathdisp {f \colon I \longrightarrow \R} { }
an infinitely often
differentiable
function,
and
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then
\mathdisp {\sum_{ k = 0}^{ \infty } \frac{ f^{( k )}(a)}{ k !} (x-a)^{ k }} { }
}
\inputfaktbeweis
{Convergent power series/R/Taylor series/Coincidence/Fact}
{Theorem}
{}
{
\faktsituation {Let \mathl{\sum _{ n= 0}^\infty c_n x^{ n }}{} denote a
power series
which
converges
on the
interval
\mathl{]-r, r[}{,} and let
\mathdisp {f \colon ]-r,r[ \longrightarrow \R} { }
denote the function defined via
Theorem 12.2
.}
\faktfolgerung {Then $f$ is infinitely often
differentiable,
and the
Taylor series
of $f$ in $0$ coincides with the given power series.}
\faktzusatz {}
}
{
That $f$ is infinitely often differentiable, follows directly from Theorem 16.1 by induction. Therefore, the Taylor series exists in particular in the point $0$. Hence, we only have to show that the $n$-th derivative has \mathl{c_n n!}{} as its value. But this follows also from Theorem 16.1 .
\inputbeispiel{}
{
We consider the function
\mathdisp {f \colon \R \longrightarrow \R
, x \longmapsto f(x)} { , }
given by
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ \defeq} { \begin{cases} 0,\, \text{ if } x \leq 0\, , \\ e^{- \frac{1}{x} },\, \text{ if } x > 0 \, . \end{cases}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
We claim that this function is infinitely often
differentiable,
which is only in $0$ not directly clear. We first show, by induction, that all derivatives of \mathl{e^{- \frac{1}{x} }}{}
have the form \mathl{p { \left( \frac{1}{x} \right) } e^{- \frac{1}{x} }}{} with certain polynomials
\mavergleichskette
{\vergleichskette
{p
}
{ \in }{\R [Z]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and that therefore the
limit
for \mathl{x \rightarrow 0,\, x >0}{} equals $0$
\zusatzklammer {see
Exercise 17.16
and
Exercise 17.17
} {} {.}
Therefore, the limit exists for all derivatives and is $0$. So all derivatives in $0$ have value $0$, and therefore the
Taylor series
in $0$ is just the
zero series.
However, the Function $f$ is in no neighborhood of $0$ the zero function, since
\mavergleichskette
{\vergleichskette
{ e^{- \frac{1}{x} }
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
}
\zwischenueberschrift{Power series ansatz}