Understanding Engineering Mathematics

9.Write down the first four terms of the sequence whosenth term is:

(i) n (ii) (n^2 + 3 n)/ 5 n (iii) cos^12 nπ (iv) ( 1 +(− 1 )n)/n

(v) 1/

√

n (vi)

(

n+sin^12 nπ

)/

( 2 n+ 1 ) (vii) arn−^1

(viii) (− 1 )nx^2 n+^1 /( 2 n+ 1 )

Investigate the limits of these sequences.

Hints (vii) limit depends onr (viii) limit depends onx

10.Write down thenth terms of the sequences

(i)

1

2

,

1

4

,

1

8

,

1

16

,... (ii) 0,

1

2

,

2

3

,

3

4

,

4

5

,... (iii)

1

1. 2

,

1

2. 3

,

1

3. 4

,...

(iv)

3

2

,

5

4

,

9

8

,

17

16

,... (v) − 1 ,

1

2

,−

1

6

,

1

24

,−

1

120

,...

(vi)

x

2. 3

,

2 x^2

3. 4

,

3 x^3

4. 5

,...(vii) −

x^3

3

,

x^5

5

,−

x^7

7

,... (viii)

x

2

,

2 x

3

,

3 x

4

,

4 x

5

,...

Investigate the limits of these sequences.

Hints(vi), (vii), and (viii) depend onx.

11.Evaluate the following limits:

(i) lim

x→∞

x^2 e−x (ii) lim

n→∞

1

n

(iii) lim

n→∞

1

n!

(iv) lim

n→∞

n

n+ 2

(v) lim

n→∞

∣

∣

∣

∣

x

n+ 1

∣

∣

∣

∣ (vi) nlim→∞

∣

∣

∣

∣

xn

n+ 1

∣

∣

∣

∣

(vii) lim

n→∞

a(l−rn)

l−r

12.Ifl=limn→∞

∣

∣

∣

∣

un+ 1

un

∣

∣

∣

∣findlin the following cases:

(i) un=

1

n^2

(ii) un=

1

n!

(iii) un=

(− 1 )n(x/ 2 )n

(n+ 1 )

13.Plot or sketch the functionf(x)= 3 x^3 − 4 x+5 and verify that it has a root near
x=− 1 .5. Use the Newton – Raphson method to obtain this root to 2 decimal places.

14.Using the Newton – Raphson method find to 2 decimal places the value of

(i)

√

291. 7 (ii)^3

√

3. 074

Hint(i) Treat as the equationx^2 − 291. 7 =0. Similarly for (ii).

15.Find the sequence of thenth partial sums for the following series, i.e. the sequence:

S 1 ,S 2 ,S 3 ,...

If possible, find a formula forSnand thus evaluate lim

n→ 8

Sn.