# MacLaurin formula

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