 |
RAN701S - REAL ANALYSIS - 1ST OPP - JUNE 2022 |
|
|
1 Page 1 |
▲back to top |
pe.
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: JUNE 2022
PAPER: THEORY
DURATION: 3 HOURS
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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.
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)
|
|
2 Page 2 |
▲back to top |
QUESTION 1
Let (x,) be a sequence of real numbers and x € R.
1.1. Define what does it mean to say the sequence (x,,) converges to x?
[2]
1.2. Use the definition in (1.1) to establish the sequence (=) converges to 0.
[9]
QUESTION 2
. . (n-vn
Find Jima. (=).
[5]
QUESTION 3
3.1. Define what does it mean to say a sequence (x,) in Ris a Cauchy sequence?
[3]
3.2. Use the defi.nitoigoene in (3.1.) to show that the sequence ( n7+=3n)\\., ) is a Cauchy sequence. [14]
QUESTION 4
4.1 Determin. e the sum of dicno=0 (aye) 6(n47) usi:ng partia. l fracti.on decompositisaos n.
[14]
4.2. Determin. e whether the seri.es }ir_,(—1)”
QUESTION 5
n37n++2e2sn+1
converges absolutely or conditai:onally.
[10]
Use the Epsilon- delta (€ — 6 ) definition to show that in
=-—1.
[13]
x7
QUESTION 6
Show, using the definition of uniform continuity, the function f(x) = = is uniformly
continuous on [0, 3].
[10]
QUESTION 7
Apply the mean value theorem to prove that |sin y— sin x| < |y — x| forallx,yER.
[7]
|
|
3 Page 3 |
▲back to top |
QUESTION 8
8.1. Find the fifth degree Taylor polynomial ps(x) of f(x) = e* centered at 0.
[7]
8.2. Determine a bound for the error when e°°approximated by ps (x).
[6]
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER