A series representation for a function f having continuous derivatives of all orders. The series is , where f(r)(0) means drf(x)/dxr evaluated at 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 ex and sin 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/x2) 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.