The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 121 Friday, March 12, 2004 12:39 PM

Principles of Calculus 121

TAYLOR SERIES EXPANSION

An important relationship used in economics and finance theory to

approximate how the value of a function, such as a price function, will

change is the Taylor series expansion. We begin by establishing Taylor’s

theorem. Consider a continuous function with continuous derivatives

up to order n in the closed interval [a,b] and differentiable with contin-

uous derivatives in the open interval (a,b) up to order n + 1. It can be

demonstrated that there exists a point ξ ∈(a,b) such that

f ′′()(ba)^2 f()n()a(ba– )n

()= fa a –

a –

fb ()+ f ′()(ba)+ ---------------------------------+ ...+ -------------------------------------+ Rn

2! n!

where the residual Rn can be written in either of the following forms:

f (n +^1 )()ξ(ba– )n +^1

Lagrange’s form: Rn = ----------------------------------------------------

(n + 1 )!

f (n +^1 )()ξ(b – ξ)n(ba– )

Cauchy’s form: Rn = --------------------------------------------------------------

n!

In general, the point ξ ∈(a,b) is different in the two forms. This

result can be written in an alternative form as follows. Suppose x and x 0

are in (a,b). Then, using Lagrange’s form of the residual, we can write

f′′()x(xx– ()x –

0 )

(^2) fn()(xx
0 )
n
fx()= fx() 0 + f′()x(xx– 0 )+ ------------------------------------+ ...+ ----------------------------------------
2! n!
f (n +^1 )()ξ(xx– 0 )n +^1

• 
  

  ---

  
  

  (n + 1 )!
  If the function f is infinitely differentiable, i.e., it admits derivatives
  of every order and if
  lim R = 0
  n → ∞ n
  the infinite series obtained is called a Taylor series expansion (or simply
  Taylor series) for f(x). If x 0 = 0, the series is called a Maclaurin series.