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A particular case of the [[Taylor formula|Taylor formula]]. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620901.png" /> have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620902.png" /> derivatives at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620903.png" />. Then in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620904.png" /> of this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620905.png" /> can be represented in the form
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<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/m/m062/m062090/m0620906.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620907.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620908.png" />-th order remainder term, can be represented in some form or other.
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A particular case of the [[Taylor formula|Taylor formula]]. Let a function  $  f $
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have  $  n $
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derivatives at  $  x = 0 $.  
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Then in some neighbourhood  $  U $
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of this point  $  f $
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can be represented in the form
  
The term "MacLaurin formula"  is also used for functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m0620909.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209010.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209011.png" /> in the MacLaurin formula is taken to be a multi-index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209012.png" /> (see [[MacLaurin series|MacLaurin series]]). The formula is named after C. MacLaurin.
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$$
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f ( x) = \
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\sum _ { k= } 0 ^ { n }
  
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\frac{f ^ { ( k) } ( 0) }{k ! }
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x  ^ {k} + r _ {n} ( x) ,\ \
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x \in U ,
 +
$$
  
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where  $  r _ {n} ( x) $,
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the  $  n $-
 +
th order remainder term, can be represented in some form or other.
 +
 +
The term  "MacLaurin formula"  is also used for functions of  $  m $
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variables  $  x = ( x _ {1} \dots x _ {m} ) $.
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In this case  $  k $
 +
in the MacLaurin formula is taken to be a multi-index,  $  k = ( k _ {1} \dots k _ {m} ) $(
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see [[MacLaurin series|MacLaurin series]]). The formula is named after C. MacLaurin.
  
 
====Comments====
 
====Comments====
For some expressions for the remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062090/m06209013.png" /> and for estimates of it see [[Taylor formula|Taylor formula]].
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For some expressions for the remainder $  r _ {n} ( x) $
 +
and for estimates of it see [[Taylor formula|Taylor formula]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR></table>

Latest revision as of 04:11, 6 June 2020


A particular case of the Taylor formula. Let a function $ f $ have $ n $ derivatives at $ x = 0 $. Then in some neighbourhood $ U $ of this point $ f $ can be represented in the form

$$ f ( x) = \ \sum _ { k= } 0 ^ { n } \frac{f ^ { ( k) } ( 0) }{k ! } x ^ {k} + r _ {n} ( x) ,\ \ x \in U , $$

where $ r _ {n} ( x) $, the $ n $- th order remainder term, can be represented in some form or other.

The term "MacLaurin formula" is also used for functions of $ m $ variables $ x = ( x _ {1} \dots x _ {m} ) $. In this case $ k $ in the MacLaurin formula is taken to be a multi-index, $ k = ( k _ {1} \dots k _ {m} ) $( see MacLaurin series). The formula is named after C. MacLaurin.

Comments

For some expressions for the remainder $ r _ {n} ( x) $ and for estimates of it see Taylor formula.

References

[a1] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108
How to Cite This Entry:
MacLaurin formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_formula&oldid=15362
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article