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(MATH202)[2006-2010](f)midterm1~=br8ltkdm^_51597.pdf
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Notations: R denotes the set of all real numbers. Q denotes the set of all rational numbers. The variable n inthe problemsbelowtakesonpositiveintegervalues1,2,3,....
1. (11 Marks) Let A be a nonempty bounded subset of R such that inf A = 1 and
sup A =3. Let
B = {.2x(15+ xy): x (2,4) Q,y A}.

Prove that B isbounded. Determine (with proof)the in.mum and supremumof B.
2. (11 Marks) Prove the sequence {xn}converges, where
7
x1 =5 and xn+1 =
xn +5,
and .nd its limit. Show work!
3. (11 marks) Do either (a) or (b)below:
(a) Determine (with proof)all positive irrationalnumbersb suchthat

cos(k.3b)
(2k.b).(ln k)2 +1.
k=1
converges.
(b) Determine (with proof)whether the set

cos(k.3b)
S = .b : b (0,+)\Q and . converges.
(2k.b).(ln k)2 +1.
k=1
is countable or not.
CEnd ofPaperC

Directions: Thisisa closedbook exam. Every student must showworkin every problem
with correct details legibly to receive marks. Answers alone are worth very little!!! Notations: R denotes the set of all real numbers. Q denotes the set of all rational numbers.
1. (a)(5marks) Determine (with proof)all nonnegativerealnumber bsuchthat the series
.
k=1 2k+3
converges. (This means for the remaining nonnegative real number b,
k2(b+1)k
you also have to explain whythe series diverges.) Show details!
(b) (5 marks) Let a1,a2,a3,... be real numbers in the open interval (0,1) such that
.
ak
converges. Determine (with proof)whether

.
sin ak
1.ak
converges or not.

k=1 k=1
2. (12 Marks) Let D be a nonempty bounded subset of R such that inf D = 3 and
sup D =5. Let
A = {xy +xy 3 : x (2,]Q,y D}.
Show that A isbounded. Determine (with proof)the in.mum and supremumof A.
3. (11 marks) Let S be the set of all points(x,y)in the coordinate plane that satisfy the equations
22
x 2 +y = a and y = x 2 .x 3 +b
for some a,b Q with a .b. Determine (with proof)if S is countable or not.

= CEnd ofPaperC
Notations: R denotes the set of all real numbers. Q denotes the set of all rational numbers.
1
1. (a) (6 marks) Determine if the series .(cos k)sin.
.converges. Show work! k2 +2
k=1
(b) (8 marks) Prove the sequence {xn} converges, where


4xn +xn x1 =1 and xn+1 = 3
and .nd its limit. Show work!
2. (a) (6 marks) Determine (with proof)the supremum and in.mum of B = .cos x +siny : x,y (0,/2] Q..
(b) (8 marks) Let D and E be nonemptybounded subsets of R suchthat
inf D =3, sup D =5, inf E =7 and sup E =9. Determine (with proof)the supremum and in.mum of the set
A = x + : x D, y E.
y


3. (5 marks) Prove that there exists a positive real number c which does not equal to

2
any numberof the form2a+b , where a,b Q.
CEnd ofPaperC

Notations: R denotes the set of all real numbers. Q denotes the set of all rational numbers.
2kk2
1. (a) (6 marks) Determine (with proof)if . converges.
(2k)!
k=1

cos k
(b) (6 marks) Determ