The Chemistry Maths Book, Second Edition

7.9 Exercises 221

EXAMPLE 7.17Differentiation and integration of power series

0 Exercises 79, 80

7.9 Exercises

Section 7.2

Find (a) the general term and (b) the recurrence relation for the sequences:

1.1, 4, 7, 10,= 2.1, 3, 9, 27,= 3.

Find the first 6 terms of the sequences:

4. 
   
   
   
   5. 
      
   
   
   
   


5. 
   
   
   
   7. 
      
   
   
   
   

8.u

n+ 2

1 = 1 u

n+ 1

1 + 12 u

n

; u

0

1 = 1 1, u

1

1 = 13 9.u

n+ 2

1 = 13 u

n+ 1

1 − 12 u

n

; u

0

1 = 1 1, u

1

1 = 1122

10.u

n+ 2

1 = 13 u

n+ 1

1 − 12 u

n

; u

0

1 = 1 u

1

Find the limitr 1 → 1 ∞for:

11. 
    
    
    
    12. 2
    
    
    
    

r

13. 
    
    
    
    14. 
        
        
        
        15. 
            
            
            
            16. 
                
            
            
            
            
        
        
        
        
    
    
    
    

17.Find the limit of the sequence{u

n+ 1

2 u

n

}foru

n+ 2

1 = 1 u

n+ 1

1 + 12 u

n

; u

0

1 = 11 , u

1

1 = 13

(see Exercise 8).

Section 7.3

Find the sum of (i)the first nterms, (ii)the first 10 terms:

18. 
    

    11 + 151 + 191 + 1131 +1- 19. 31 − 121 − 171 − 1121 −1- 20. 11 + 131 + 191 + 1271 +1-

    
    


19. 
    

Find the sum of the first nterms:

22.x

3

1 + 1 x

5

1 + 1 x

7

1 +1- 23.x 1 + 12 x

2

1 + 14 x

3

1 +1-

1

1

3

1

9

1

27

+++ +

331

561

2

2

rr

rr

++

−−

r

rr

2

++ 1

r

r+ 2

1

r+ 2

1

3

r

w

w

w

n

n

n

+

==

11

u 1;

xx

x...

x

=

• 
  

=, , ,

1

2

123

()

;

v

n

n

= n...













=,,,

2

3

; 012

uu u

rr+

=+ =

11

1

2

0;

1

1

5

1

25

1

125

,− , ,− ,...

=++++++=− −+cx

xxxx

xc

2345

2345

 ln( ) 1

ZZ

1

1

1

234

−













= +++++

x

dx ()x x x x dx

=+ + + + =

−

12 3 4

1

1

23

2

xx x

x



()

d

dx x

d

dx

xx x x

1

1

1

234

−











= +++++()