Padmashree Dr DY Patil Vidyapeeth 2007 B.E Computer Science Mathamtics - Question Paper
Sunday, 20 January 2013 06:30Web
M.Sc. DEGREE EXAMINATION, 2007
( MATHEMATICS )
( 1st YEAR )
( PAPER - II )
120. REAL ANALYSIS
( Revised Regulations )
May ] [ Time : three Hours
Maximum : 100 Marks
PART ? A (8 × five = 40)
ans any 8 ques..
All ques. carry equal marks.
1. State and prove generalized Mean ? Value
theorem.
2. If f 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 fÎÂ(a), prove that for every t > 0
there exists a partition P such that
U (P, f, a) ? L (P, f, a ) < t.
5. provide 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 f is of bounded variation [a, b], prove that
f (b) ? f (a) = Pab ? Nab.
9. If (1 + an) converges absolutely,
prove that it converges.
10. obtain the value of (1 ? n?2).
PART ? B (3 × 20 = 60)
ans any 3 ques..
All ques. carry equal marks.
11. (a) State and prove chain rule for differentiation.
(10)
(b) Prove that f is of bounded variation on
[a, b] iff f can be expressed as the difference
of 2 increasing functions. (10)
12. Let a be of bounded variation on [a, b].
Let V(n) be the total variation of a on
[a, x] and let V(a) = 0. If f ÎÂ(a),
prove that f ÎÂ (V) ? (20)
13. (a) If fn Î Â(a) for every n, if fn ® f uniformly
on [a, b] and if gn(n) = fn (t) da(t),
prove that f ÎÂ(a) and gn ® g uniformly
where g(n) = f(t) da (t) (10)
(b) State and prove Bernstein theorem. (10)
14. (a) State and prove Littlewood?s 3rd principle.
(10)
(b) Prove that every absolutely continuous
function is the indefinite integral of its
derivative. (10)
15. (a) If an ³ 0 n, prove that p (1?an)
converges iff the series å an converges.
(10)
(b) State and prove Cauchy condition for
infinite product. (10)
Earning: Approval pending. |