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RAN701S - REAL ANALYSIS - 2ND OPP - JULY 2023 |
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nAm I BIA UnlVERSITY
OF SCIEnCE AnD TECHnDLOGY
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:
JULY 2023
3 HOURS
PAPER:THEORY
MARKS: 100
SUPPLEMENTARY/SECONDOPPORTUNITYEXAMINATION QUESTION PAPER
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 [16]
Use the definition of the limit of a sequence to establish the following limits.
= 1.1. nl-i+moo ( 3,/ n+l ) 0.
[8]
1.2.
If lim Cxn) = 3, then
= lim
( 3xn-
1
)
2.
[8]
n->oo
n->oo
4
QUESTION 2 [14]
Determine whether each of the following sequences converges or diverges.
[8]
[6]
QUESTION 3 [10]
= Prove that lim Cxn)= 0 if and only if lim (lxnD 0. Give an example to show that the
convergence of (lxnD need not imply the convergence of Cxn).
QUESTION 4 [15]
4.1. Determine whether the series L~=:l: converges or diverges.
[7]
4.2. Determine
whether
the
L~- series
-
1
(-l)nn
3n+4
converges absolutely or conditionally.
[8]
Question 5 [10]
(.Jn) Show that the sequence (xn) =
is a Cauchy sequence.
[10]
QUESTION 6 [9]
6.1. Define what does it mean to say a sequence Cxn)in lffi.is increasing?
[2]
.Jz 6.2. Let x1 = 1, Xn+1= + Xn for n E M. Show that Cxn)is increasing.
[7]
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QUESTION 7 [18)
Let Ai;;; JR{and let f: A R
7.1. Define what does it mean to say f is uniformly continuous on A?
[3]
= 7.2. Let f(x) x 2 •
(a) Use the definition of uniform continuity to show that f is uniformly continuous on [-4, 2].
[7]
(b) Use the nonuniform criteria to show that f is not uniformly continuous on (-oo, oo). [8]
QUESTION 8 [8]
Apply the mean value theorem to prove that lln y - In xi $ 4ly - xi for x < y and
X ,y E [41 , 4].
ENED OF SUPPLEMENTARY/ SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
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