Real Analysis Answers

6. a) By showing that the remainder after n -terms tends to zero, find Maclaurin’s series
expansion of sin 2x . (5)
b) Find the greatest value of the function 7 ( ) 2 3 4
4 3 2
f x = x − x − x + x + over the
interval ]1,0[

5. a) Show that the local maximum value of
x
x





 1
is e
e
/1
. (4)
b) Verify Cauchy Mean Value Theorem for the functions
, ,1[ ]4
1
( ) = , ( ) = x∈
x
f x x g x . (3)
c) Show that [ 1+ x ≤ e , ∀x ∈ ,0[ ∞
x
. Does the inequality hold for x < 0 ? Justify your
answer.

4. a) Using the principle of mathematical induction, prove that 7 is a factor of
+ ∀ ∈ N
− +
n
n n
3 2 ,
2 1 1
. (4)
b) Show that the equation 0 2 5 12 3 2
x − x + x − = has a root which is a positive real
number. (4)
c) Prove that the set 





, K
8
3
,
7
3
,
6
3
is a countable set.

3. a) Explain the order completeness property of R , and use it to show that the set






∈
+
= n N
n
n
S
1
has a supremum as well as infimum in R . (3)

b) Let f be the function defined by








+ ∈ ∞
∈
−
− ∈ ∞
=
1( 2 ) , if ,2[ [
, if ,1[ [2
3 2
2 ,1 if ] [1,
( )
2
2
x x
x
x
x
x x
f x
Discuss the continuity of f on [ ]− ∞, ∞ . (4)
c) Check whether the following sets are open, closed or neither:
i) ]6 ,1] [5 ∪ ,3[
ii) 




 ∪
7
10
,
4
3
,
9
5
]1,0[
iii) {5 n : n∈ N}

2. a) Prove that the sequence } { n
a where 2 2
2
3
2
+
=
n
an
, converges to 0 . (3)
b) Find the following limit, if it exists:
3
3 3
0 1 cos
sin lim
x
x x
x→ −
(2)
c) Test the convergence of the following series. (5)
i) 2 2
+ 2 2
+ 2 2
+L
7 .8
6.5
5 .6
4.3
3 .4
2.1
ii) ∑
+ − −
n
n 1 n 1

a) Prove that the sequence } { n
a where 2 2
2
3
2
+
=
n
an
, converges to 0 . (3)
b) Find the following limit, if it exists:
3
3 3
0 1 cos
sin lim
x
x x
x→ −
(2)
c) Test the convergence of the following series. (5)
i) 2 2
+ 2 2
+ 2 2
+L
7 .8
6.5
5 .6
4.3
3 .4
2.1
ii) ∑
+ − −
n
n 1 n 1

Which of the following statements are true or false? Give reasons for your answers. (10)
a) The singleton set {x} for any x ∈ R is an open set.
(b) The series is − + − +L
7
1
5
1
3
1
1 is a convergent series.
c) The function



=
+ ≠
=
−
,1 when 0
, when 0
( )
x
e e x
f x
x x
is continuous on ]1,0[ .
d) The function f defined by f (x) = | x − |2 ∀x ∈ R has a critical point at x = 2 .
e) If a function has finitely many points of discontinuities, then the function is not
integrable.

Which of the following statements are true or false? Give reasons for your answers. (10)
a) The singleton set {x} for any xÎ R is an open set.
(b) The series is − + − +L
7
1
5
1
3
1
1 is a convergent series.
c) The function

 
=
+ ¹
=
−
1 , when 0
, when 0
( )
x
e e x
f x
x x
is continuous on [0, 1] .
d) The function f defined by f (x) = | x − 2 |"xÎ R has a critical point at x = 2 .
e) If a function has finitely many points of discontinuities, then the function is not
integrable.
2. a) Prove that the sequence { } n a where 2 2
2
3
2
+
=
n
an , converges to 0 . (3)
b) Find the following limit, if it exists:
3
3 3
0 1 cos
sin
lim
x
x x
x® −
(2)
c) Test the convergence of the following series. (5)
i) + + +L
2 2 2 2 2 2 7 .8
5.6
5 .6
3.4
3 .4
1.2
ii)  + − −
n
n 1 n 1 2 2

1. Which of the following statements are true or false? Give reasons for your answers. (10)
a) The singleton set {x} for any xÎ R is an open set.
(b) The series is − + − +L
7
1
5
1
3
1
1 is a convergent series