maclaurin series polynomial

An example of Maclaurin series

One of representations of a function as an infinite sum of terms calculated from the values of the function’s derivatives at a single point is known as a Taylor series ¹ and a Maclaurin series is a Taylor series expansion of a function about 0 ². Both series are found similar to a power series ³.

How a Maclaurin series can represent a polynomial function is given here.

Suppose there is polynomial function represented in the form of ⁴

$$\tag{1} f(x) = \sum_{i=0}^n a_i x^i$$

with $n$ is the degree of a polynomial. And the Maclaurin series is

$$\tag{2} g(x) = \sum_{i=0}^n \frac{1}{i!} \left. \frac{d^i f(x)}{dx^i} \right|_{x=0} x^i.$$

The $g(x)$ should be able to represent $f(x)$ with all of its terms. Eqn (1) will become

$$\tag{3} f^m(0) = m! \ a_m,$$

from $m$-th derivative of a polynomial at $x=0$. Substitute Eqn (3) into Eqn (2) will give

$$g(x) = \sum_{i=0}^n \frac{1}{i!} \ ( i! \ a_i ) \ x^i = \sum_{i=0}^n a_i x^i,$$

which is back to Eqn (1). It has been shown that Maclaurin series can represent a polynomial function.