1. No category

COMSATS Institute of Information Technology, Islamabad
Department of Mathematics
Assignment # 2
Program: BSM
Semester: VI
Date: Sep 22, 2015
Instructor: Dr. Muhammad Saeed Akram
Maximum Marks: 10
Deadline : Sep 30, 2015
Assignment Topics: Uniform convergence of sequence of functions.
Discuss the uniform convergence of the sequence of function {fn }, where
1. FA11-BSM-012
x
1. fn (x) =
(x ∈ R, n = 1, 2, 3, . . . ).
n
2. fn (x) =
sin nx
1 + nx
(x ∈ [0, ∞), n = 1, 2, 3, . . . ).
2. FA12-BSM-001
1. fn (x) = xn
2. fn (x) =
(x ∈ R, n = 1, 2, 3, . . . ).
xn
1 + xn
(x ∈ [0, ∞), n = 1, 2, 3, . . . ).
3. FA12-BSM-010
1. fn (x) =
x2 + nx
n
(x ∈ R, n = 1, 2, 3, . . . ).
2 x2
2. fn (x) = n3/2 xe−n
(x ∈ [−1, 1], n = 1, 2, 3, . . . ).
4. FA12-BSM-013
1
1. fn (x) = ( ) sin(nx + n)
n
2. fn (x) = (cos x)n
(x ∈ R, n = 1, 2, 3, . . . ).
(x ∈ [0, π], n = 1, 2, 3, . . . ).
5. FA12-BSM-017
1. fn (x) =
1
nx
(x ∈ (0, 1], n = 1, 2, 3, . . . ).
2 x2
2. fn (x) = nxe−n
6. FA12-BSM-021
x
1. fn (x) =
1 + nx
(x ∈ R, n = 1, 2, 3, . . . ).
(x ∈ [0, ∞), n = 1, 2, 3, . . . ).
1
−x
xe n
2. fn (x) =
n
(x ∈ (0, ∞), n = 1, 2, 3, . . . ).
7. FA12-BSM-022
1. fn (x) = xn
(x ∈ [0, 1], n = 1, 2, 3, . . . ).
2. fn (x) = (1 − |x|)n
(x ∈ (−1, 1), n = 1, 2, 3, . . . ).
8. FA12-BSM-024
1. fn (x) = n2 x(1 − x2 )n )
2. fn (x) =
x
1 + nx2
(0 ≤ x ≤ 1, n = 1, 2, 3, . . . ).
(x ∈ R, n = 1, 2, 3, . . . ).
9. SP13-BSM-001
nx
1. fn (x) =
1 + n2 x2
(x ∈ [0, ∞), n = 1, 2, 3, . . . ).
2. fn (x) = n2 xn (1 − x))
(x ∈ [0, 1], n = 1, 2, 3, . . . ).
10. SP13-BSM-002
(
1, −n ≤ x ≤ n
1. fn (x) =
0, otherwise
2. fn (x) =
x2 + nx
n
(x ∈ R, n = 1, 2, 3, . . . ).
11. SP13-BSM-006
(
n, x ∈ (0, n1 )
1. fn (x)(x) =
0, otherwise
2. fn (x) =
x2n
1 + x2n
nx
1 + nx
13. SP13-BSM-009
x
1. fn (x) = 1 −
n
2. fn (x) = (sin x)n
(x ∈ [0, 1], n = 1, 2, 3, . . . ).
(x ∈ R, n = 1, 2, 3, . . . ).
12. SP13-BSM-008
(
n, x ∈ (− n1 , n1 )
1. fn (x)(x) =
0, otherwise
2. fn (x) =
(x ∈ R, n = 1, 2, 3, . . . ).
(x ∈ [−1, 1], n = 1, 2, 3, . . . ).
(x ∈ R, n = 1, 2, 3, . . . ).
(x ∈ [0, 1], n = 1, 2, 3, . . . ).
(x ∈ [0, π], n = 1, 2, 3, . . . ).
2