Analysis 1 (MATS2101)
BSc, Physics - PHYS
Semester: First Semester
Level: 200
Year: 2016
REPUBLIC OF CAMEROON THE UNIVERSITY OF BAMENDA
Peace-Work-Fatherland P.O. Box 30 Bambili
Faculty of Science Dept. of Maths. and Comp. Sciences Lecturer(s): Dr. KAMEUGNE R.
Level: 1 Semester:
1
Academic Year: 2016/2017
C. Code: MATS2101 C.Title: Analysis
1 Duration
: 3
hours
Instructions: Answer all questions.
FIRST SEMESTER EXAMINATION
Exercise 1 (15 marks) Consider the set A = {x ; (5x - 6)
2
< 18}.
1. Prove that 2 is not a rational number. 6 marks
2. Deduce that
is not a rational number. 4 marks
3. Show that the set A has no supremum in .
Exercise 2 (40 marks) Let (u
n
) and (v
n
) be two reals sequence.
1. Define: (u
n
) converge to ; (u
n
) is a Cauchy sequence; (u
n
) and (v
n
) are adjacent sequences; (v
n
) is a
subsequence of (u
n
). 8 marks
2. Let (u
n
) and (v
n
) be two real sequences such that (u
n
) is bounded and (v
n
) converge to 0. Show that the
sequence (u
n
v
n
) converge to 0. 4 marks
3. Application: Consider the sequence w
n
=
)
(a) Prove by definition that the sequence v
n
=
converge to 0. 4 marks
(b) Deduce the convergence of (w
n
).
4- Let (
) be a subsequence of (u
n
) where : is a strictly, increasing function.
(a) Show that: for all n , > n. 3 marks
(b) Prove that (u
n
) converge to imply (
) converge to the same limit. 5 marks
(c) Give a counter example of sequence which diverge but have convergent subsequence. 2 marks
5. Suppose and are real numbers such that 0 Let u
1
= , v
1
= and u
n+1
=
!
"
#
"
and
$
%
&
'()
(a) Show that (u
n
) is a monotonically decreasing sequence and that is bounded below by 4 marks
(h) Show that (v
n
) is a monotonically increasing sequence and that is bounded above by 3marks
(c) Show that 0 –
*+,
"-.
for n 3 marks
(d) Deduce that (u
n
) and (v
n
) are convergent and have the same limit. 3 marks
6. Cauchy sequence: Let (u
n
) the sequence define by u
n
=
/
for all n * .
(a) Show that every convergence sequence is a Cauchy sequence. 4 marks
(b) Show that for all n * , u
2
n
— u
n
0
3 marks
(c) Deduce that (u
n
) is not a Cauchy sequence. 2 marks
Exercise 3 (15 marks) Consider the sequence a
n
=
1. Determine the reals and such that for all n * ,
$
*
,
4marks
2. Let s
n
be the sequence of partial sum, s
n
=
1
2
3
34
n *
(a) Evaluate S
2
, .S
3
and S
5
. 5marks
(b) Express s
n
as a function of n.
4
marks
(c) Deduce the sum {a
n
} of the series of general term (a
n
). 2marks
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