## Quick Reference

A series representation for a function f having continuous derivatives of all orders. The series is , where f^{(r)}(0) means d* ^{r}*f(

*x*)/d

*x*

*evaluated at*

^{r}*x*=0. The series may converge for all values of

*x*, or for |

*x*|<

*R*, for some positive

*R*, or it may only converge when

*x*=0. If f(

*x*) is a polynomial then the series is finite and the sum is f(

*x*). For most practical cases, if the series converges then its sum is f(

*x*)—this is true for e

*and sin*

^{x}*x*, when the series is convergent for all

*x*, and for (1+

*x*)

^{1/2}and ln(1+

*x*), when the series is convergent for |

*x*|<1. In such cases the first (

*n*+1) terms give a polynomial approximation g(

*x*) to f(

*x*), which has the property that at

*x*=0, the first

*n*derivatives of g(

*x*) and f(

*x*) are equal. There are however non-zero functions f such that f(

*x*) and all its derivatives are zero at

*x*=0, and in this case the Maclaurin series vanishes—for example, f(

*x*)=exp (−1/

*x*

^{2}) for

*x*≠0, with f(0)=0.

The Maclaurin series is of use in the theoretical development of moment-generating functions and probability-generating functions. It is the particular case of a Taylor series when *a*=0.

*Subjects:*
Probability and Statistics.

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