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Difference between revisions of "MacLaurin formula"

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$$  
 
$$  
f ( x)  = \  
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f ( x)  = \sum_{k=0}^{n}  
\sum _ { k= } 0 ^ { n }  
 
  
 
\frac{f ^ { ( k) } ( 0) }{k ! }
 
\frac{f ^ { ( k) } ( 0) }{k ! }
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====Comments====
 
====Comments====
 
For some expressions for the remainder  $  r _ {n} ( x) $
 
For some expressions for the remainder  $  r _ {n} ( x) $
and for estimates of it see [[Taylor formula|Taylor formula]].
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and for estimates of it see [[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>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin, "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR>
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</table>

Latest revision as of 10:42, 20 January 2024


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=55236
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article