Difference between revisions of "MacLaurin formula"
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$$ | $$ | ||
| − | f ( x) = \ | + | f ( x) = \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 [[ | + | and for estimates of it see [[Taylor formula]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, | + | <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 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 |
MacLaurin formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=MacLaurin_formula&oldid=47742