A
Number and Algebra
8
Maclaurin’s series
8.1 Introduction
Some mathematical functions may be represented as
power series, containing terms in ascending powers
of the variable. For example,
ex= 1 +x+x^2
2!+x^3
3!+···sinx=x−x^3
3!+x^5
5!−x^7
7!+···and coshx= 1 +
x^2
2!+x^4
4!+···(as introduced in Chapter 5)Using a series, calledMaclaurin’s series, mixed
functions containing, say, algebraic, trigonometric
and exponential functions, may be expressed solely
as algebraic functions, and differentiation and inte-
gration can often be more readily performed.
8.2 Derivation of Maclaurin’s theorem
Let the power series forf(x)be
f(x)=a 0 +a 1 x+a 2 x^2 +a 3 x^3 +a 4 x^4
+a 5 x^5 +··· (1)wherea 0 ,a 1 ,a 2 ,...are constants.
Whenx=0,f( 0 )=a 0.
Differentiating equation (1) with respect toxgives:
f′(x)=a 1 + 2 a 2 x+ 3 a 3 x^2 + 4 a 4 x^3
+ 5 a 5 x^4 +···(2)Whenx=0,f′( 0 )=a 1.
Differentiating equation (2) with respect toxgives:
f′′(x)= 2 a 2 +(3)(2)a 3 x+(4)(3)a 4 x^2+(5)(4)a 5 x^3 +··· (3)Whenx=0,f′′(0)= 2 a 2 = 2 !a 2 , i.e.a 2 =
f′′(0)
2!Differentiating equation (3) with respect toxgives:f′′′(x)=(3)(2)a 3 +(4)(3)(2)a 4 x+(5)(4)(3)a 5 x^2 +··· (4)Whenx=0,f′′′(0)=(3)(2)a 3 = 3 !a 3 , i.e.a 3 =f′′′(0)
3!Continuing the same procedure givesa 4 =fiv(0)
4!,a 5 =fv(0)
5!, and so on.Substituting fora 0 ,a 1 ,a 2 ,...in equation (1) gives:f(x)=f(0)+f′(0)x+f′′(0)
2!x^2+f′′′(0)
3!x^3 +···i.e.f(x)=f( 0 )+xf′(0)+x^2
2!f′′(0)+x^3
3!f′′′(0)+···(5)Equation (5) is a mathematical statement called
Maclaurin’s theoremorMaclaurin’s series.8.3 Conditions of Maclaurin’s series
Maclaurin’s series may be used to represent any
function, sayf(x), as a power series provided that
atx=0 the following three conditions are met:(a)f( 0 )=∞
For example, for the function f(x)=cosx,
f(0)=cos 0=1, thus cosxmeets the condition.
However, if f(x)=lnx, f(0)=ln 0=−∞,
thus lnxdoes not meet this condition.
(b) f′(0),f′′(0),f′′′(0),...=∞
For example, for the function f(x)=cosx,
f′(0)=−sin 0=0,f′′(0)=−cos 0=−1, and so