Download OU B.Sc 2019 Dec 3rd Sem (2nd Year) 8073E Mathematics Question Paper

Download OU (Osmania University) B.Sc (Bachelor of Science) 2019 Dec 3rd Sem (2nd Year) 8073E Mathematics Previous Question Paper

Code NO. 80'?. I E
FACULTY OF SCIENCE ?
B.Sc. lIl-Semester (CBCS) Examination, November! December 2019
ematics (Real Analysis)
Subject : Math 036
Time ' 3 Hours Paper .- Ill ( ) Max. Marks: 80 ,
? ? PART?A(5x4=20Mafks) , . 1, _._
(Short Answer Type) _ 1 300! \g" H .?
Note : Answer any FIVE of the following questions. 6 ? wt!?
'3 Find gi?snmhere s" = n3 +1 -n.
2, Ptg?e'?that every convergent sequence is a Cauchy sequence. ?
ence {an} where ?FsmF?EJ
3, Pindthe set of subsequential limits, of the sequ
4 Test the convergence of the series 2 ?1'; 3 .
' t
3'
5- Find the interval of convergence of the series 2%. ?
6? De?ne the uniform convergence of-a sequence of fugctionst
' 7 ?t is a bounded function on [a, b], prove that L gum under usual notations
} ,8 Prove that every continuous function f on [gab] i '?h?t?grable.
' PART - B (45353??? ,.__r-?3rks)
(Essay WM" Type)
Note: Answer ALL the following queg?ons.
9 ,ra)P1:ove that:
l
(i) 1min") = (ii) lim{ *jr??Tfor a > 0'.
H 5330: +1? 3
OR
(b) Let (3") be a seq??hge in R. If lim sn is de?ned (as a real number or + van or wan),
then prove that Iim sup 5n = lim 5n = lim inf s... ?
10 (a) If (Sn). cdagerges to a positive rea! number s and (tn) is any sequeneemen
9 that imp sweat.1 = 5. lim sup tn. " ? ? ' H ? ?
0R
(b) ?State1?nd prove the comparison test.
11 (a) gnaw that if the sen?es 29., conve?ges uniformly mamsm
ygsupu am I: xe S} = e. ' " '
_ . , 7 108
(:5) W .120?) = H?x?? 4) ?at 3 51196.; IL 1 "9m
mums uni?eimly oni?, 13;, ? ? ? 7'