Chapter 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!

+···

andcoshx= 1 +

x^2

2!

+

x^4

4!

+···

(as introduced in Chapter 5)

Using a series, calledMaclaurin’s series, mixed func-
tions containing, say, algebraic, trigonometric and expo-
nential functions, may be expressed solely as algebraic
functions, and differentiation and integration can often
be more readily performed.
To expand a function using Maclaurin’s theorem,
some knowledge of differentiation is needed (More on
differentiationis given in Chapter 27). Here is a revision

yorf(x) dxdyorf′(x)

axn anxn−^1

sinax acosax

cosax −asinax

eax aeax

lnax^1 x

sinhax acoshax

coshax asinhax

of derivatives of the main functions needed in this

chapter.

Given a general function f(x),thenf′(x)is the

first derivative, f′′(x)is the second derivative, and so

on. Also,f( 0 )means the value of the function when

x= 0 ,f′( 0 )means the value of the first derivative when

x=0, and so on.

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!