Anna University Chennai 2007 M.Sc Computer Science Mathamatics - Question Paper
Mathamatics
(b) State and prove Bernstein theorem. (10)
4
14. (a) State and prove Littlewoods third principle.
(b) Prove that every absolutely continuous function is the indefinite integral of its derivative. (10)
15. (a) If a >0 V , prove that 71 (1-a )
\ j n n v ir
converges iff the series a converges.
(10)
(b) State and prove Cauchy condition for infinite product. (10)
Name of the Candidate :
7 7 0 1 M.Sc. DEGREE EXAMINATION, 2007
( MATHEMATICS )
( FIRST YEAR )
( PAPER - II )
120. REAL ANALYSIS
( Revised Regulations )
May ] [ Time : 3 Hours
Maximum : 100 Marks
PART-A (8x5 = 40)
Answer any EIGHT questions.
All questions carry equal marks.
1. State and prove generalized Mean - Value theorem.
2. If / and g are of functions of bounded variation on [a, b], prove that f + g is of bounded variation.
3. If P' is a refinement of P, Prove that
U(P',f,a )<U(p, f, a)
4. if fe 9t(a), prove that for each t > 0 there exists a partition P such that
U (P, f, a) - L (P, f, a ) < t.
5. Give an example to show that a sequence of continuous functions need not converge to a continuous function.
6. State and prove Weierstrass M-test.
7. Prove that [0, 1] is uncountable.
8. If / is of bounded variation [a, b], prove that
f (b) - f (a) = Pab - Nab.
9. If "K (1 + an) converges absolutely, prove that it converges.
a
10. Find the value of (1-n-2).
n = 2
Answer any THREE questions.
All questions carry equal marks.
11. (a) State and prove chain rule for differentiation.
(b) Prove that / is of bounded variation on [a, b] iff / can be expressed as the difference of two increasing functions. (10)
12. Let a be of bounded variation on [a, b]. Let V(n) be the total variation of OC on [a, x] and let V(a) = 0. If / e 9(0C), prove that / e 9 (V) ? (20)
13. (a) If fn e 9(a) for each n, if fn~>f uniformly
x
on [a, b] and if g (n) = | f (t) da(t),
a
prove that f e 9(0C) and g g uniformly x
where g(n) = j f(t) da (t) (10)
a
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