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UNIVERSITY OF MALTAFACULTY OF SCIENCE

DEPARTMENT OF MATHEMATICS

Engineering-1 Problem Sheet 4

(1) Using the theorems on limits of sequences and the definition ofa limit of a sequence, find the limit as n , if any, of thefollowing sequences:

sin n

n,

4 2n3n + 2

,n4 + 1

n6, 2

1

n ,n4 + 1

n2.

(2) Show that the sequence an

= n+2n+1

is monotonic decreasing andbounded below, and hence show that it converges. Prove thatthe limit is 1.

(3) Show that the sequence an

= 1nn2+1

is monotonic increasing (forn = 3, 4, 5, . . . ) and bounded above, and hence show that itconverges. Prove that the limit is 0.

(4) The sequence un is defined by the recursion formula un+1 =3u

n, u1 = 1. Prove that the sequence is monotonic increasing

and bounded above, and hence show that it converges. Provethat the limit is 3.

(5) Using the Ratio test, determine the convergence or otherwise ofthe following series:

(i)n=1

(n + 1)3

2n, (ii)

n=1

n2

3n, (iii)

n=1

nn

n!.

(You may assume that limn

1 +1

n

n= e.)

(6) Using the integral test, determine the convergence or otherwiseof the following series:

(i)n=1

1

(n + 2)3, (ii)

n=1

1

n2.

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(7) Consider the infinite series

1 + 2r + r2 + 2r3 + r4 + 2r5 + . . . ,

where (a) r = 23

, (b) r = 23

, (c) r = 43

. Show that the ratiotest is inapplicable for the above series and using the nth roottest, show that the series converges for cases (a) and (b), anddiverges in case (c).

(8) Using the theorems on limits of functions, prove the limits

(i) limxa

x2 a2

x2 + 2ax + a2= 0, a = 0;

(ii) limx0

x2 a2

x2 + 2ax + a2= 1, a = 0.

(9) By finding the limits

limx0+

|x|x

and limx0

|x|x

show that

limx0

|x|x

does not exist.

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