## Taylor series

Problems such as maximization of a function f or finding the root of an equation

f (x)= 0 arise frequently in economics. However, often the function may be difficult to work with, motivating the use of approximations to the function f that are

easier to study and provide insight regarding f . One relatively simple class of

approximating functions are the polynomials. A function h is a polynomial in x if

it has the form:

h(x)= a0+a1x+a2x2+·· ·+anxn,

where each ai is a constant. So, h is the sum of terms where each term is a

constant multiplied by x to the power of some integer, such as a3x 3, where a3

is the constant and x is raised to the power of 3. The function h is called an nth

order polynomial because the highest power is n (assuming an 6= 0). For example, h(x)= 2+x+5x2+8x3 is a third-order polynomial, as is g(x)= 2x3. Under suitable circumstances, a large class of functions can be approximated by a polynomial, so

that given the function f , one may find a polynomial such as h approximating f .

This makes it possible to discover properties of f by study of the simpler func-

tion h and that fact is a primary motivation for studying these approximations.

Taylor’s theorem concerns the approximation of functions with polynomials and

provides the theoretical basis for the polynomial approximations.