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