B.Sc. (Hons.) Math IV (Fourth) Semester
Examination 2011-12
Course Code:BAS402
Paper ID: 0984106
Real Analysis
Time: 3 Hours
Max. Marks: 70
Note: Attempt six questions in all. Q. No. 1 is compulsory.
1.
Attempt any five of the following (limit your answer to 50
words).
(4x5=20)
a)
 1
Show that the sequence {S n } where S n  1  
 n
n
b)
c)
d)
e)
f)
n
, is
 1
convergent and that lim 1   lies between 2 and 3.
 n
A necessary condition for convergence of an infinite, series
lim
u n  0.
u n is that
n
If f is continuous at x  x0 where f ( x0 )  0, then a positive
number  can be found such that f (x) has the same sign as
f ( x0 ) for every value of x in ]x0   , x0   [.
Show that the function f(x) is not differentiable at x=0 and
x=1.
Verify Rolle’s theorem in the case of the functions (any one).
i)
f ( x)  2 x 3  x 2  4 x  2.
f ( x)  sin x in [0, π].
ii)
iii)
f ( x)  ( x  a) m ( x  b) n , where m and n are positive
integers, and x lies in the interval [a,b].
If a function f(x) is (i) continuous in a closed internal [a, b],
and (ii) differentiable in the open interval ]a, b[i.e. a < x < b,
then there exists at least one value ‘c’ of x lying in the open
f (b)  f (a)
 f (c).
interval ]a, b[ such that.
ba
n=1, 2, 3, ….. If <fn> converges uniformly to the function f
on [a,b], then f  R [a, b] and
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g) Letf ( x)  x on [0,1],

0
x dx and
 x dx by dissecting [0,1] into n equal
b
a
fn ( x) dx.
a
0
parts and hence show that f  R [0,1].
h) If lim S n  l and lim t n  l , then lim ( S n t n )  l l . .
6. a)
2. a) Every convergent sequence is bounded.
(5)
b) If {an }, {bn } be two sequences such that lim an  a,
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lim bn  b, then
i)
lim ( an  bn )  lim an  lim bn  a  b.
ii)
lim ( an bn )  (lim an )(lim bn )  ab.
iii)
lim ( an / bn )  (lim an / lim bn ) if b  0, bn  0  n .
3. a) Show that the series
1.2
3.4
5.6
 2 2  2 2  ................ . Converges.
2 2
3 .4
5 .6
7 .8
b) Test the convergence of the series:
1
 n11 / n .
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4. a) A necessary and sufficient condition for the integrability of a
bounded function f is that to every   0, there corresponding
a   0 such that for every partition P with norm µ( P )   ,
u ( P, f )  L ( P, f )   .
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b) Every continuous function is integrable.
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nx
0  x  1 and n  1,2,3....
when
1 n2 x2
examine as to whether the sequence  f n  is uniformly
convergent on R.
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b) Let  f n  be a sequence of real valued functions defined on
5. a)
lim
 f ( x) dx  n   
1
1
Calculate
b
Let f n ( x) 
the closed and bounded interval [a,b] and let f n  [a, b] for
b)
A function f(x) is defined as follows
( x 2 / a )  a, when x  a

f(x) = 
0,
when x  a
a  (a 2 / x), when x  a

Prove that the function f (x) is continuous at x=a.
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If a function f(x) is continuous in a closed interval [a,b], then
it is bounded in that interval.
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7. a)
b)
Every bounded sequence has at least one limit point.
Every absolutely convergent series is convergent but
converse is not necessarily true i.e. convergence need
imply absolute convergence.
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the
not
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8.
If function f is bounded and integrable and admits of a
primitive f on [a,b], then
b
 f dx  f (b)  f (a).
a
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