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