Download OU B.Sc 2019 Dec 3rd Sem (2nd Year) 8073E Mathematics Question Paper

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FACULTY OF SCIENCE | December 2019

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

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Mathematics (Real Analysis)

Subject:

Time : 3 Hours Max. Marks: 80

PART-A (5x4=20 Marks)

(Short Answer Type)

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Note : Answer any FIVE of the following questions.

1. Find lim s_n, where s_n = v(n + 1) - vn.
2. Prove that every convergent sequence is a Cauchy sequence.
3. Find the set of subsequential limits of the sequence {a_n} where a_n = sin(np/3).
4. Test the convergence of the series S (1/n³).

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5. Find the interval of convergence of the series S xⁿ/n.
6. Define the uniform convergence of a sequence of functions.
7. If f is a bounded function on [a, b], prove that L(f) = U(f) under usual notations.
8. Prove that every continuous function f on [a, b] is integrable.

PART-B (4x15=60 Marks)

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(Essay Answer Type)

Note: Answer ALL the following questions.

1. (a) Prove that:
   1. lim (n^1/n) = 1
   2. lim (a^1/n) = 1 for a > 0.
   OR (b) Let (s_n) be a sequence in R. If lim s_n is defined (as a real number or +8 or -8), then prove that lim sup s_n = lim s_n = lim inf s_n.

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2. (a) If (s_n) converges to a positive real number s and (t_n) is any sequence then prove that lim sup (s_nt_n) = s * lim sup t_n. OR (b) State and prove the comparison test.
3. (a) Show that if the series Sg_n converges uniformly on a set S, then lim sup(|g_n(x)| : x ? S) = 0. OR (b) Let f_n(x) = n²xⁿ(1-x) for x ? [0, 1]. Then prove that the sequence converges uniformly on [0, 1].
4. (a) Prove that a bounded function f on [a, b] is integrable on [a, b] if and only if for each e > 0 there exists a partition P of [a, b].