The Chemistry Maths Book, Second Edition

7.3 Finite series 201

are important in combinatorial theory (Section 21.6) and are used in a popular deriva-

tion of the Boltzmann distribution in statistical thermodynamics (Example 21.12).

EXAMPLE 7.5Expand the trinomial(a 1 + 1 b 1 + 1 c)

3

.

The possible values of (n

1

, n

2

, n

3

)are

(3, 0, 0), (0, 3, 0), (0, 0, 3), (2, 1, 0), (2, 0, 1), (1, 2, 0), (1, 0, 2),

(0, 2, 1), (0, 1, 2), (1, 1, 1)

with distinct multinomial coefficients

Therefore

(a 1 + 1 b 1 + 1 c)

3

1 = 1 (a

3

1 + 1 b

3

1 + 1 c

3

) 1 + 1 3(a

2

b 1 + 1 a

2

c 1 + 1 ab

2

1 + 1 ac

2

1 + 1 b

2

c 1 + 1 bc

2

) 1 + 16 abc

0 Exercises 34, 35

The method of differences

Many simple finite series can be summed if the general term u

r

can be written as the

difference

u

r

1 = 1 v

r

1 − 1 v

r− 1

Then

= 1 (v

1

1 + 1 v

2

1 +1-1+ 1 v

n− 1

1 + 1 v

n

) 1 − 1 (v

0

1 + 1 v

1

1 +1-1+ 1 v

n− 1

)

= 1 v

n

1 − 1 v

0

EXAMPLE 7.6Find the sum of the series.

The general term can be written as

and, therefore,

1

1

11

rr()+ r r 1

=−

• 
  

r

n

rr nn

=

∑

• 
  

=

⋅

• 
  

⋅

++

• 
  

1

1

1

1

12

1

23

1

() () 1



r

n

r

r

n

r

r

n

r

u

===

−

∑∑∑

=−

111

1

vv

3

300

1

3

210

3

3

111

6

!

!!!

,

!

!!!

,

!

!!!

===