Math 2111 Advanced Calculus (I)
Homework 14
Hand in Problems: All Problems
1. In the Dini’s Theorem (see Theorem 5.19), if the following conditions hold (i) K is compact.
(ii) {fn} is a sequence of continuous function on K.
(iii) {fn} converges to a continuous function f pointwise on K.
(iv) fn(x) ≤ fn+1(x) for all x ∈ K and n ∈ N then {fn} converges uniformly to f on K.
(a) Give a counterexample if only (i)(ii)(iii) hold.
(b) Give a counterexample if only (ii)(iii)(iv) hold.
2. Suppose f (x) =
∞
X
n=1
fn(x) and g(x) =
∞
X
m=1
gm(x) converges uniformly on E, and f and g are bounded on E.
(a) Prove that
f (x)g(x) =
∞
X
n=1
∞
X
m=1
fn(x)gm(x) =
∞
X
m=1
∞
X
n=1
fn(x)gm(x) =
∞
X
m,n=1
fn(x)gm(x)
converges uniformly on E.
(b) Prove that h(x) =
∞
X
n=1
sin nx
2n converges uniformly on [0, 2π].
(c) Evaluate Z 2π
0
h(x)2 dx.
3. Let {fn}∞n=1 be a sequence of continuous real-valued functions on [0, ∞) which satisfy (i) fn(x) ≥ 0 for every x ∈ [0, ∞) and n ∈ N.
(ii) fn(x) ≤ fn+1(x) for every x ∈ [0, ∞) and n ∈ N.
(iii) {fn}∞n=1 converges to a continuous f pointwise on [0, ∞).
Prove that
n→∞lim Z ∞
0
fn(x) dx = Z ∞
0
f (x) dx.
(Note: In this problem, we allow the case that those improper integrals could be equal to infinity.)
4. Suppose that, for some c 6= 0, the series
∞
X
n=0
ancn converges. Prove that the power series
∞
X
n=0
anxn
converges uniformly on [−b, b] for any 0 < b < |c|.
Lecture Note:
• (Page 198) 1. Problem 5.7 2. Problem 5.8 3. Problem 5.9(1)(3)
