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Download OU B.Sc 2018 Dec 2nd Year 3072 Real Analysis Question Paper

Download OU (Osmania University) BSc (Bachelor of Science - Maths, Electronics, Statistics, Computer Science, Biochemistry, Chemistry & Biotechnology) 2nd Year 1st Semester (Third Semester) (2-1) 3072 Real Analysis Previous Question Paper

This post was last modified on 06 February 2020

OU B-Sc Last 10 Years 2010-2020 Question Papers || Osmania University


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Code No. 3072/E

FACULTY OF SCIENCE

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B.Sc. III-Semester (CBCS) Examination, November / December 2018

Subject : Mathematics

Paper - II : Real Analysis (DSC)

Time : 3 Hours Max. Marks: 80

PART ~ A (5 x 4 = 20 Marks)

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(Short Answer Type)

Note : Answer any FIVE of the following

  1. Determine the limit of the sequence {sn}, where sn = v(n2 + 1) - n
  2. Let t1 = 1 and tn+1 = v(1 + 2tn) for n = 1. Find the lim tn
  3. Let an = n sin(1/n), then find lim sup an and lim inf an.
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  5. Show that ?n=28 1/(n(log n)p) converges if and only if p > 1.
  6. For n = 0, 1, 2, 3,... let an = (4 + (-1)n)/n. Find lim sup (an)1/n and lim inf (an)1/n.

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Code No. 3072/E

  1. (a)
    1. If the sequence (sn) converges, then prove that every subsequence converges to the same limit.
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    3. State and prove Bolzano-Weierstrass theorem.
    OR (b) If (sn) converges to a positive real number s and (tn) is any sequence then prove that lim sup sn tn = s lim sup tn.
  2. (a) Let (fn) be a sequence of functions defined and uniformly Cauchy on a set S ? R. Then prove that there exists a function f on S such that fn ? f uniformly on S. OR (b) Derive an explicit formula for ?n=18 xn for |x| < 1 and hence evaluate ?n=18 n2xn-1.
  3. (a) Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b] and P ? Q, then prove that L(f, P) = L(f, Q) = U(f, Q) = U(f, P). OR (b) Prove that a bounded function f on [a, b] is Riemann integrable on [a, b] if and only if it is Darboux integrable, in which case the values of the integrals agree.

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