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RAN701S - REAL ANALYSIS - 2ND OPP - JULY 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSOC; 07BAMS
LEVEL: 7
COURSE CODE: RAN701S
COURSE NAME: REAL ANALYSIS
SESSION: JULY 2022
PAPER: THEORY
DURATION: 3 HOURS
MARKS: 100
SUPPLEMENTARY /SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
DR NEGA CHERE
MODERATOR:
PROF FORTUNE MASSAMBA
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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QUESTION 1
Use the Epsilon- delta (€ — 6) definition of convergence of a sequence to show that
(=)2 converges to 2.
[3]
QUESTION 2
Find
lim
n-oo
(soot
ent)
vn+2
[3]
QUESTION 3
3.1. Show directly from the definition that if (x,,) and (y,) are Cauchy sequences, then
(Xy — Yn) is a Cauchy sequence.
[7]
3.2. Prove that a convergent sequence is a Cauchy sequence.
[3]
QUESTION 4
Let x, = 2 and forn > 1,let x,4, =4- = Assuming that (x,) converges, find
lim (x,).
[8]
QUESTION 5
_ (23 5.1. Determine whether the sequencXe= (
_45 67) i
ore ) converges or diverges.[8]
(—1)"2"n2
5.2. Determine whether the series })r~-9
converges conditionally or absolutely? [10]
n!
QUESTION 6
Use Epsilon- delta (€ — 6) definition to show that Jim = = —2.
[13]
QUESTION 7
LetA& Rand let f:A > R.
7.1. Define what does it mean to say f is uniformly continuous on A?
[3]
7.2. Use the definition in (5.1) to show that f(x) = x? is uniformly on [—2, 2].
[10]
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QUESTION 8
8.1. Find the fourth Taylor Polynomial centered at 0 for the function f(x) = —
[10]
8.2. Apply the mean value theorem to prove that |In y—Inx| < 4y —x| for’ x<y<4.
[7]
END OF SUPPLEMENTAY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER