Answer to Question #58696 in Real Analysis for esita

Question

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