Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 12
Exercise 1 Study the convergence (pointwise, uniform) of the following sequences of functions:
fn(x) = xn(1 −xn), gn(x) = xn(1 −x)n
on the interval [0,1].
Exercise 2 Study the convergence (pointwise, uniform) of the following sequence of functions:
fn(x) = nx
1 + n2x2
on the interval [0,+∞).
Exercise 3 Let (fn)be the sequence of functions defined by fn(x) = 1
nif |x| ≤ nand by fn(x)=0if
|x|> n.
(1) Prove that fnconverges uniformally to the zero function 0.
(2) Prove that
lim
n→∞ Z∞
−∞
fn(x)dx 6=Z∞
−∞
limn→∞fn(x)dx.
Does it contradict the theorem on convergnece-integration ?
Exercise 3 Let (fn)be the sequence of functions defined by fn(x) = nx2
1+nx if x≥0and by fn(x) =
nx3
1+nx2if x < 0.
(1) Prove that fnis differentiable and that f0
nis continuous.
(2) Find fsuch that fnconverges pointwise to f.
(3) Prove that fnconverges uniformally to f.
(4) Find gsuch that f0
nconverges pointwise to g.
(5) Does f0
nconverges uniformally to g?
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