The Chemistry Maths Book, Second Edition

222 Chapter 7Sequences and series

Use equation (7.11) to expand in powers of x:

24.(1 1 + 1 x)

5

25.(1 1 + 1 x)

7

Calculate the binomial coefficients , r 1 = 1 0, 1, =, n, for

26.n 1 = 13 27.n 1 = 14 28.n 1 = 17

Use equation (7.13) or (7.14) to expand in powers of x:

29.(1 1 − 1 x)

3

30.(1 1 + 13 x)

4

31.(1 1 − 14 x)

5

32.(3 1 − 12 x)

4

33.(3 1 + 1 x)

6

34. (i) Calculate the distinct trinomial coefficients .(ii) Use the coefficients to

expand(a 1 + 1 b 1 + 1 c)

4

.

35. (i) Calculate the distinct coefficients .(ii) Use the coefficients to expand

(a 1 + 1 b 1 + 1 c 1 + 1 d)

3

.

36.Find.

37. (i)Verify that , then (ii)find the sum of the series.


38. (i)Express in partial fractions, then (ii)show that


39. (i)Verify that(1 1 + 1 r)

3

1 − 1 r

3

1 = 13 r

2

1 + 13 r 1 + 11 , then (ii)show that

40. (i)Expand(1 1 + 1 r)

6

1 − 1 r

6

, then (ii)use the series in Table 7.1 to find the sum of the series

.

Section 7.4

(i)Expand in powers of xto terms inx

6

. (ii)Find the values of xfor which the series converge:
41. 42. 43.
44. (i) Use the geometric series to express the number 12 (10

6

1 − 1 1)as a decimal fraction.

(ii)Show that the decimal representation of 1 2 7 can be written as 1428572 (10

6

1 − 1 1)

(see Section 1.4).

45.The vibrational partition function of a harmonic oscillator is given by the series

whereθ

v

1 = 1 hν

e

2 kis the vibrational temperature. Confirm that the series is a convergent

geometric series, and find its sum.

qe

n

nT

v

v

=

=

∑

−/

0

∞

θ

1

2 +x

1

15

2

• x

1

13 − x

r

n

r

=

∑

1

5

r

n

rnn n

=

∑

=++

1

2

1

6

()( ) 12 1

r

n

rr r n n

=

∑

++

=−

++

1

1

12

1

4

1

()( ) ()( )21 2

1

rr r()( )++ 12

r

n

rr

=

∑

• 
  

1

1

() 2

1

2

1

2

11

rr()+ r r 2

=−

• 
  













n

nn

=

∑

• 
  

1

10

1

() 1

3

1234

!

!!!!

nnnn

4

123

!

nnn!!!

n

r















