The Chemistry Maths Book, Second Edition

7.6 MacLaurin and Taylor series 209

(ii) The coefficient of x

n

in the series

isc

n

1 = 1 (−3)

n

2 n

2

. By the ratio test,|c

n

2 c

n+ 1

| 1 = 1 (1 2 3)(n 2 n 1 + 1 1)

2

1 → 1 (1 2 3)asn 1 → 1 ∞

and the radius of convergence isR 1 = 1123. The series therefore converges when

|x| 1 < 1123. It also converges when|x| 1 = 1123 (see Example 7.8).

0 Exercises 52–57

The MacLaurin series

10

We saw in Section 7.4 that the geometric series can be regarded as the expansion of the

function 12 (1 1 − 1 x)in powers of x. Similarly, the exponential series can be regarded as

the expansion of that functionf(x) 1 = 1 e

x

whose derivative is equal to itself,f′(x) 1 = 1 f(x).

Many other functions can be expanded in this way.

Letf(x) be a function of xthat can be represented as a power series

f(x) 1 = 1 c

0

1 + 1 c

1

x 1 + 1 c

2

x

2

1 + 1 c

3

x

3

1 + 1 c

4

x

4

1 +1- (7.19)

The coefficientsc

0

, c

1

, c

2

,1=can be obtained in the following way. The derivatives of

the function are

f′(x)

f′′(x)

f′′′(x)

• 
  

Then, lettingx 1 = 10 ,

f(0) 1 = 1 c

0

, f′(0) 1 = 1 c

1

, f′′(0) 1 = 1 2!c

2

, f′′′(0) 1 = 1 3!c

3

, =

and, in general, for the nth derivative,

f

dfx

dx

nc c

n

f

n

x

n

n

nn

() (n

()

()

0

1

0

=

















=! , =

!

=

))

()0

== + +

df

dx

cx cx

3

3

3

2

4

624 

==++ +

df

dx

ccxcx

2

2

23 4

2

26 12 

==+ + + +

df

dx

ccxcx cx

12 3

2

4

3

23 4

()−

=

∑

3

2

1

x

n

n

n

∞

10

Colin MacLaurin (1698–1746), professor of mathematics at Edinburgh. The series called after him appeared

in his Treatise of fluxions(1742), but the more general Taylor series was published in 1715, and was known to

the Scottish mathematician James Gregory (1638–1675). The Treatisecontains also the method of deciding the

maximum 2 minimum question by investigating the sign of a higher derivative.