Understanding Engineering Mathematics

3.3.9 Finite series

(i) a=1, r=

1

2

S 6 = 2

(

1 −

1

26

)

=

63

32

(ii) a= 0 .1, r= 0. 1 S 6 =

1

9

(

1 −

1

106

)

= 0. 1111111

(iii) a=2, r= 2 S 8 = 2 ( 28 − 1 )= 510

(iv) a=

1

3

, r=

1

3

S 4 =

1

2

(

1 −

1

34

)

=

40

81

(v) a=1, r= 0. 1 S 5 =

10

9

(

1 −

1

105

)

= 1. 1111

3.3.10 Infinite series

A.S∞=

a

1 −r

(i) No

(ii) No

(iii) Yes, diverges to∞

(iv) No

(v) Yes

(

a= 1 ,r=^13

)

S∞=^32

(vi) No

(vii) Yes(a= 2 ,r=− 2 )diverges

(viii) Yes(a= 0. 2 ,r= 0. 2 )S∞=^14

B. (i) ∞ (ii) 4 (iii) diverges to−∞

(iv)^32

3.3.11 Infinite binomial series

(i) 1+(− 1 )x+

(− 1 )(− 2 )(x)^2

2!

+

(− 1 )(− 2 )(− 3 )(x)^3

3!

= 1 −x+x^2 −x^3

(ii) 1+(− 1 )(− 3 x)+

(− 1 )(− 2 )(− 3 x)^2

2!

+(− 1 )(− 2 )(− 3 )(− 3 x)^3

3!

= 1 + 3 x+ 9 x^2 + 27 x^3