The Chemistry Maths Book, Second Edition

198 Chapter 7Sequences and series

This equation can be regarded in two ways:

(i) the value of the sum(1 1 + 1 x 1 + 1 x

2

1 +1-1+ 1 x

n− 1

)is(1 1 − 1 x

n

) 2 (1 1 − 1 x),

(ii) the series(1 1 + 1 x 1 + 1 x

2

1 +1-1+ 1 x

n− 1

)is the expansionof the function(1 1 − 1 x

n

) 2 (1 1 − 1 x)

in powers of x.

This concept of the expansion of one function in terms of a set of (other) functions

provides an important tool for the representation of complicated (or unknown)

functions in the physical sciences.

0 Exercises 20–23

The binomial expansion

The binomial expansion is the expansion of the function(1 1 + 1 x)

n

in powers of xwhen

nis a positive integer. Examples of such expansions are

(1 1 + 1 x)

2

1 = 111 + 12 x 1 + 1 x

2

(1 1 + 1 x)

3

1 = 111 + 13 x 1 + 13 x

2

1 + 1 x

3

(1 1 + 1 x)

4

1 = 111 + 14 x 1 + 16 x

2

1 + 14 x

3

1 + 1 x

4

In the general case,

(7.11)

with general term

EXAMPLE 7.3Expand(1 1 + 1 x)

6

in powers of x.

By equation (7.11), withn 1 = 16 ,

0 Exercises 24, 25

=++++++1 6 15 20 15 6

23456

xx x xxx

• 
  

×× ××

××××

• 
  

×× ×× ×

×× ×× ×

65432

54321

654321

65432

5

x

11

6

x

()116

65

21

654

321

6543

4

623

+=++

×

×

• 
  

××

××

• 
  

×× ×

×

xxx x

3321

4

××

x

nn n ...n r

r

x

r

()( )( )

!

−− −+ 12 1

()

() ()( )

11

1

2

12

3

23

+=++

−

!

• 
  

−−

!

xnx ++

nn

x

nn n

xx

nn