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''of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923101.png" />, for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923102.png" /> that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923103.png" /> times differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923104.png" />''
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''of degree $n$, for a function $f$ that is $n$ times differentiable at $x=x_0$''
  
 
The polynomial
 
The polynomial
 +
$$P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k.$$
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The values of the Taylor polynomial and of its derivatives up to order $n$ inclusive at the point $x=x_0$ coincide with the values of the function and of its corresponding derivatives at the same point:
 +
$$f^{(k)}(x_0)=P_n^{(k)}(x_0),\quad k=0,\dotsc,n.$$
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The Taylor polynomial is the best polynomial approximation of the function $f$ as $x\to x_0$, in the sense that
  
<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/t092310/t0923105.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$f(x)-P_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0,$</td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
The values of the Taylor polynomial and of its derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923106.png" /> inclusive at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923107.png" /> coincide with the values of the function and of its corresponding derivatives at the same point:
 
 
 
<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/t092310/t0923108.png" /></td> </tr></table>
 
 
 
The Taylor polynomial is the best polynomial approximation of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t0923109.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231010.png" />, in the sense that
 
 
 
<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/t092310/t09231011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
 
and if some polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231012.png" /> of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231013.png" /> has the property that
 
 
 
<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/t092310/t09231014.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231015.png" />, then it coincides with the Taylor polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231016.png" />. In other words, the polynomial having the property (*) is unique.
 
 
 
If at least one of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231018.png" />, is not equal to 0 at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092310/t09231019.png" />, then the Taylor polynomial is the principal part of the [[Taylor formula|Taylor formula]].
 
  
 +
and if some polynomial $Q_n(x)$ of degree not exceeding $n$ has the property that
 +
$$f(x)-Q_n(x)=o\left((x-x_0)^m\right),\quad x\to x_0,$$
 +
where $m\ge n$, then it coincides with the Taylor polynomial $P_n(x)$. In other words, the polynomial having the property (*) is unique.
  
 +
If at least one of the derivatives $f^{(k)}(x)$, $k=0,\dotsc,n$, is not equal to $0$ at the point $x_0$, then the Taylor polynomial is the principal part of the [[Taylor formula]].
  
 
====Comments====
 
====Comments====
 
For references see [[Taylor formula|Taylor formula]].
 
For references see [[Taylor formula|Taylor formula]].

Revision as of 16:53, 16 March 2013

of degree $n$, for a function $f$ that is $n$ times differentiable at $x=x_0$

The polynomial $$P_n(x)=\sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k.$$ The values of the Taylor polynomial and of its derivatives up to order $n$ inclusive at the point $x=x_0$ coincide with the values of the function and of its corresponding derivatives at the same point: $$f^{(k)}(x_0)=P_n^{(k)}(x_0),\quad k=0,\dotsc,n.$$ The Taylor polynomial is the best polynomial approximation of the function $f$ as $x\to x_0$, in the sense that

$f(x)-P_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0,$ (*)

and if some polynomial $Q_n(x)$ of degree not exceeding $n$ has the property that $$f(x)-Q_n(x)=o\left((x-x_0)^m\right),\quad x\to x_0,$$ where $m\ge n$, then it coincides with the Taylor polynomial $P_n(x)$. In other words, the polynomial having the property (*) is unique.

If at least one of the derivatives $f^{(k)}(x)$, $k=0,\dotsc,n$, is not equal to $0$ at the point $x_0$, then the Taylor polynomial is the principal part of the Taylor formula.

Comments

For references see Taylor formula.

How to Cite This Entry:
Taylor polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Taylor_polynomial&oldid=12991
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article