Karnataka State Open University (KSOU) 2011-1st Sem M.Sc Mathematics (Real Analysis) - Question Paper
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I Semester M.Sc. Examination, May 2011 MATHEMATICS Real Analysis - I
Max. Marks : 80
Time : 3 Hours
Note x I) Answer any fixe questions.
2) All questions carry equal marks.
1. a) Prove the following statements in the field of real numbers
(2+1+2+1)
i) If x + y = x + z , then y = z
ii) If x + y = x, then y = 0
iii) If x + y = 0, then y = -x
iv) - (-x) = x.
(5 c 5)
b) Prove that Q is dense in R and R/Q is also dense in R.
2. a) Define a countable set. Show that the set W of all integers is countable.
Further, show that every subset of a countable set is also countable. 8
b) Define a limit point and closed set in a metric space. Show that every point of an open interval (a, b) in R' is a limit point. Further, show that the closed interval (a, b) is a closed set in R'. 8
3. a) In a metric space show that closed subsets of a compact set are also compact. 5
b) Prove that the closed interval (a, b) is compact in R'. 6
4. a) Prove that limit of a convergent sequence is unique. 4
b) If lim xn = x and lim y = y, then show that lim x y =xy. 4
n-co n oo n n go
c) Define x1 = 1, xn +1 = + n>\. V
1 n 3 + 2xn
Find x2, x3 and x4. Show that {xn} is monotonically increasing and bounded sequence. Compute the limit of the sequence.
v n j
b) Prove that a sequence of real numbers is convergent if and only if it is a Cauchy sequence.
c) Show that if xn = , then lim xn = e11.
n!
00 ]
6. a) Prove that Z ~r converses if p > I and diverses if p < 1. 8
n=0 np
b) Investigate the behavior of the following series :
i)
n=l
(4+4)
ii)
n=0
f 1 '\ ' n +1 '
2n +1
7. a) Prove that if Z an converges and if {bn} is monotonic and bounded, then
Z an bn converges. 5
b) State Kummers test for convergence or divergence of a series. Deduce Ratio test and Raabes test from Kummers test. 6
00 Y
c) Discuss the convergence or divergence of S n0gnp . 5
8. a) State and prove Mertens theorem for Cauchy product of two series. 10
b) If D an is a series of complex numbers which converges absolutely, then
prove that every rearrangement of 2 an converges. 6
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