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A representation of a function as a sum of its [[Taylor polynomial|Taylor polynomial]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923001.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923002.png" />) and a remainder term. If a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923003.png" /> of one variable is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923004.png" /> times differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923005.png" />, its Taylor formula has the form
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A representation of a function as a sum of its [[Taylor polynomial|Taylor polynomial]] of degree $n$ ($n=0,1,2,\dotsc$) and a remainder term. If a real-valued function $f$ of one variable is $n$ times differentiable at a point $x_0$, its Taylor formula has the form
 
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$$f(x)=P_n(x)+r_n(x),$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923006.png" /></td> </tr></table>
 
 
 
 
where
 
where
 
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$$P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923007.png" /></td> </tr></table>
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is its [[Taylor polynomial]], while the remainder term $r_n(x)$ can be written in Peano's form:
 
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$$r_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0.$$
is its [[Taylor polynomial|Taylor polynomial]], while the remainder term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923008.png" /> can be written in Peano's form:
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If the function $f$ is $n+1$ times differentiable in some neighbourhood $(x_0-\delta,x_0+\delta)$, $\delta>0$, of a point $x_0$, then in this neighbourhood the remainder term can be written in the Schlömilch–Roch form
 
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$$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0)))}{n!p}(1-\theta)^{n-p+1}(x-x_0)^{n+1},$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t0923009.png" /></td> </tr></table>
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where $p=1,\dotsc,n+1$; as special cases there are the Lagrange form
 
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$$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{(n+1)!}(x-x_0)^{n+1}$$
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230010.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230011.png" /> times differentiable in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230013.png" />, of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230014.png" />, then in this neighbourhood the remainder term can be written in the Schlömilch–Roch form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230015.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230016.png" />; as special cases there are the Lagrange form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230017.png" /></td> </tr></table>
 
 
 
 
and the Cauchy form
 
and the Cauchy form
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$$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{n!}(1-\theta)^n(x-x_0)^{n+1},$$
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$$0<\theta<1,\quad x\in(x_0-\delta,x_0+\delta);$$
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the number $\theta$ depends on $x$, $p$ and $n$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230018.png" /></td> </tr></table>
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If the derivative of order $n+1$ of the function $f$ is integrable on the interval with end points $x$ and $x_0$, then the remainder term can be written in integral form:
 
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$$r_n(x)=\frac{1}{n!}\int_{x_0}^x f^{(n+1)}(t)(x-t)^n dt.$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230019.png" /></td> </tr></table>
 
 
 
the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230020.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230023.png" />.
 
 
 
If the derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230024.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230025.png" /> is integrable on the interval with end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230027.png" />, then the remainder term can be written in integral form:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092300/t09230028.png" /></td> </tr></table>
 
  
 
Taylor's formula, with all forms of the remainder term given above, can be extended to the case of a function of several variables. Taylor's formula is also valid for mappings of subsets of a normed space into similar spaces, and in this case the remainder term can be written in Peano's form or in integral form.
 
Taylor's formula, with all forms of the remainder term given above, can be extended to the case of a function of several variables. Taylor's formula is also valid for mappings of subsets of a normed space into similar spaces, and in this case the remainder term can be written in Peano's form or in integral form.
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====Comments====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>
 
 
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>
 

Latest revision as of 11:04, 27 December 2013

A representation of a function as a sum of its Taylor polynomial of degree $n$ ($n=0,1,2,\dotsc$) and a remainder term. If a real-valued function $f$ of one variable is $n$ times differentiable at a point $x_0$, its Taylor formula has the form $$f(x)=P_n(x)+r_n(x),$$ where $$P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$$ is its Taylor polynomial, while the remainder term $r_n(x)$ can be written in Peano's form: $$r_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0.$$ If the function $f$ is $n+1$ times differentiable in some neighbourhood $(x_0-\delta,x_0+\delta)$, $\delta>0$, of a point $x_0$, then in this neighbourhood the remainder term can be written in the Schlömilch–Roch form $$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0)))}{n!p}(1-\theta)^{n-p+1}(x-x_0)^{n+1},$$ where $p=1,\dotsc,n+1$; as special cases there are the Lagrange form $$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{(n+1)!}(x-x_0)^{n+1}$$ and the Cauchy form $$r_n(x)=\frac{f^{(n+1)}(x_0+\theta(x-x_0))}{n!}(1-\theta)^n(x-x_0)^{n+1},$$ $$0<\theta<1,\quad x\in(x_0-\delta,x_0+\delta);$$ the number $\theta$ depends on $x$, $p$ and $n$.

If the derivative of order $n+1$ of the function $f$ is integrable on the interval with end points $x$ and $x_0$, then the remainder term can be written in integral form: $$r_n(x)=\frac{1}{n!}\int_{x_0}^x f^{(n+1)}(t)(x-t)^n dt.$$

Taylor's formula, with all forms of the remainder term given above, can be extended to the case of a function of several variables. Taylor's formula is also valid for mappings of subsets of a normed space into similar spaces, and in this case the remainder term can be written in Peano's form or in integral form.

Taylor's formula allows one to reduce the study of a number of properties of a function differentiable a specified number of times to the substantially simpler problems of studying these properties for the corresponding Taylor polynomial. This is the basis of various and numerous applications of the Taylor polynomial, for instance to the computation of limits of functions, to the investigation of their extreme points, their points of inflection, intervals of convexity and concavity, as well as to the convergence of series and integrals and to estimates of the speed of their convergence or divergence.

References

[1] V.A. Il'in, V.A. Sadovnichii, B.Kh. Sendov, "Mathematical analysis" , Moscow (1979) (In Russian)
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)


[a1] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1957)
[a2] W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24
[a3] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78
[a4] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Taylor formula. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Taylor_formula&oldid=16172
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article