Understanding Engineering Mathematics

(vii)a,ar,ar^2 ,ar^3 , (divergent if|r|>1, limit=aifr=1, convergent to 0 if

|r|<1 and oscillating ifr=− 1 )

(viii)−

x^3

3

,

x^5

5

,−

x^7

7

,

x^9

9

, (limit=0if|x|<1 and divergent otherwise)

10. (i)

1

2 n

,( 0 ) (ii)

n− 1

n

,( 1 ) (iii)

1

n(n+ 1 )

,( 0 )

(iv)

2 n+ 1

2 n

,( 1 ) (v)

(− 1 )n

n!

,( 0 )

(vi)

nxn

(n+ 1 )(n+ 2 )

,(0if|x|= 1 ,divergent otherwise)

(vii) (− 1 )n

x^2 n+^1

2 n+ 1

,(0if|x|= 1 , divergent otherwise)

(viii)

nx

n+ 1

,(x)

11. (i) 0 (ii) 0 (iii) 0 (iv) 1 (v) 0 (vi) |x|

(vii)

a

1 −r

if|r|<1, divergent otherwise

12.(i) 1 (ii) 0 (iii)

∣

∣

∣

x

2

∣

∣

∣

13.−1.55

14.(i) 17.08 (ii) 1.45

15. (i) Sn= 2 −

(

1

2

)n− 1

,lim

n→∞

Sn= 2

(ii) Sn=

1

4

[

3 −

(

−

1

3

)n− 1 ]

,lim

n→∞

Sn=

3

4

(iii) Sn=n^2 ,limn→∞Sn=∞

(iv) No simple expression forSn, but in fact lim

n→∞

Sn=e

(v) No simple expression forSn, but in fact lim

n→∞

Sn=∞

16. (i) 6

[

1 −

( 1

3

)n]

(C) (ii) 2n(n− 4 ) (D) (iii) 2n−1(D)

(iv)

[

1 −

(x

2

)n]/(

1 −

x

2

)

(C for|x|<2)

17. (i) (− 1 )n (D) (ii) 1. 01 ( 1. 01 )n−^1 (D) (iii) (. 99 )n+^1 (C)

(iv)

1

( 49 +n)

(D) (v)

107 −n

n

(C) (vi)

(− 1 )n+^1

10 (n+ 3 )

(C)

(vii) n^2

(

3

4

)n

(C) (viii)

n

n+ 1

(D) (ix) (. 2 )n−^1 (C)