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
- Determine the limit of the sequence {sn}, where sn = v(n2 + 1) - n
- Let t1 = 1 and tn+1 = v(1 + 2tn) for n = 1. Find the lim tn
- Let an = n sin(1/n), then find lim sup an and lim inf an.
- Show that ?n=28 1/(n(log n)p) converges if and only if p > 1.
- 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
- (a)
- If the sequence (sn) converges, then prove that every subsequence converges to the same limit.
- State and prove Bolzano-Weierstrass theorem.
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- (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.
- (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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This download link is referred from the post: OU B-Sc Last 10 Years 2010-2020 Question Papers || Osmania University
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