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RAN701S - REAL ANALYSIS - 1S OPP - JUNE 2023 |
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nAmlBIA UnlVERSITY
OF SCIEnCE AnD TECHnOLOGY
FACULTYOF HEALTH,NATURALRESOURCEASND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS,STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION:Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATIONCODE: 07BAMS
LEVEL: 7
COURSECODE: RAN701S
COURSENAME: REALANALYSIS
SESSION:
DURATION:
JUNE 2023
3 HOURS
PAPER:THEORY
MARKS: 100
FIRSTOPPORTUNITYEXAMINATION QUESTIONPAPER
EXAMINER
DR. NA CHERE
MODERATOR:
PROF. F MASSAMBA
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. Number the answers clearly.
4. All written work must be done in blue or black ink and sketches
must be done in pencil.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
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QUESTION 1 [11]
Let (xn) be a sequence of real numbers and x E lit
1.1. Define what does it mean to say the sequence Cxn)converges to x?
[2]
(n~n 1.2. Use the definition
in part (1.1) to establish the sequence
n
32
+Zn)
converges
to
-3.
[9]
QUESTION2 [13]
Determine whether each of the following sequences is convergent or divergent.
[7]
[6]
QUESTION3 [10]
3.1. Define what does it mean to say a sequence Cxn)in JR{is bounded?
[3]
3.2. Prove that if Cxn)is convergent then it is bounded.
(7]
Question 4 [13]
4.1. Define what does it mean to say a sequence (xn) in JR{is a Cauchy sequence?
[3]
4.2. Show that the sequence (-2nn- -2) is a Cauchy sequence.
(10]
QUESTION 5 [16]
5.1. Find the sum of the series I:::'=Con+2;Cn+3)1 if it converges.
[8]
I:::'=o 5.2. Determine whether the series
(-1)"4"n
n!
2
converges absolutely or conditionally.
[8]
QUESTION6 [13]
= Use the Epsilon- delta (E,8) definition to show that lim 3x+s 2.
X->1 X+3
1
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QUESTION 7 [16]
= - 7.1. Use the definition of uniform continuity to show that the function f(x)
1
x+-l
is uniformly
continuous on [O, oo).
[9]
(t) = 7.2. Use the nonuniform continuity criterion to show that the function f(x) sin
is not
uniformly continuous on (0, oo).
[7]
QUESTION 8 [8]
Apply the mean value theorem to prove that Itan y - tan xi ::; 2 ly - xi for x < y and
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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