The Chemistry Maths Book, Second Edition

7.6 MacLaurin and Taylor series 211

with general term

The series (7.21) is called the binomial series, and is the generalization of the

binomial expansion, equation (7.11), to arbitrary powers. In fact, when ais a positive

integer n, the (n 1 + 1 1)th derivative and all higher derivatives are zero, and (7.21)

reduces to the binomial expansion. When ais not a positive integer, the series has

radius of convergenceR 1 = 11. Thus, applying the ratio test,

and the binomial series converges for values− 11 < 1 x 1 < 1 ± 1. It may also converge when

x 1 = 1 ±1.

The more general form, the expansion of (x 1 + 1 y)

a

, is obtained from (7.21) by

factorizing out the termx

a

if|x| 1 > 1 |y|ory

a

if|y| 1 > 1 |x|. Thus, if|x| 1 > 1 |y|,

(7.22)

and the series converges since|y 2 x| 1 < 11. Equation (7.22) can also be written as

(7.23)

EXAMPLES 7.12

(i)

= 111 + 1 x 1 + 1 x

2

1 + 1 x

3

1 +1-

(ii)

=+ − + − 1 +

2816

5

128

23 4

xx x x



() ()

!

11 ()

1

22

12

1

2

1

2 2

1

2

+=+













• 
  

()

−

()

• 
  

()

−

xxx

11

2

3

2 3

3

()

−

()

• 
  

!

()x 

1

1

111

12

2

1

12

−

=− =+−−+

−−

−+

−

−

x

() ()()xxx

()()

!

()

( ))( )( )

!

()

−−

−+

23

3

3

x 

()

() ()( )

xy x ax y

aa

xy

aa a

aa a a

+=+ +

−

!

• 
  

−−

−− 122

1

2

12

33

33

!

• 
  

−

xy

a



=+













• 
  

−

!













• 
  

−−

xa

y

x

aa y

x

aa a

a

1

1

2

12

2

() ()())

3

3

!













• 
  

















y

x



()xy x

y

x

aa

a

+= +













1

R

c

c

r

ar

rr

r

r

==

−

=

→→

+

lim lim

∞∞

1

1

aa a ...a n

n

x

n

()( )( )−− −+

!

12 1