Anna University Chennai 2007 M.Sc Computer Science Mathamatics - Question Paper

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) = P_a^b - N_a^b.

9.    If "K (1 + a_n) 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.

(10)

(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 f_n e 9(a) for each n, if f_n~>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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