# Difference between revisions of "MacLaurin formula"

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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