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

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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 {s_n}, where s_n = v(n² + 1) - n
2. Let t₁ = 1 and t_n+1 = v(1 + 2t_n) for n = 1. Find the lim t_n
3. Let a_n = n sin(1/n), then find lim sup a_n and lim inf a_n.

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4. Show that ?_n=2⁸ 1/(n(log n)^p) converges if and only if p > 1.
5. For n = 0, 1, 2, 3,... let a_n = (4 + (-1)ⁿ)/n. Find lim sup (a_n)^1/n and lim inf (a_n)^1/n.

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Code No. 3072/E

10. (a)
    1. If the sequence (s_n) converges, then prove that every subsequence converges to the same limit.

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    2. State and prove Bolzano-Weierstrass theorem.
    OR (b) If (s_n) converges to a positive real number s and (t_n) is any sequence then prove that lim sup s_n t_n = s lim sup t_n.
11. (a) Let (f_n) 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 f_n ? f uniformly on S. OR (b) Derive an explicit formula for ?_n=1⁸ xⁿ for |x| < 1 and hence evaluate ?_n=1⁸ n²x^n-1.
12. (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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