Math Problem Statement
Over the given interval
I
determine the sets of uniform convergence for:
f
n
(x)=
x
1+n
x
2
,I=R.
Hint: To find the suprimum of
g
n
(x)=|
f
n
(x)−f(x)|
on
R,
let
t=|x|
for
t≥0.
Another trick, let
u=
n
−
−
√
|x|
and use the inequality
u
1+
u
2
≤
1
2
for all
u∈R.
Question 2
Select one:
a.
f
⟶
uniform
0
on
R.
b.
f
⟶
uniform
0
on
[−1,1].
c.
f
⟶
uniform
0
on
[0,1].
d.
f
⟶
uniform
0
on
Solution
We are asked to determine the sets of uniform convergence for the sequence of functions over the interval . The goal is to identify the interval on which converges uniformly to 0.
Step 1: Pointwise Convergence
First, let's determine the pointwise limit of as .
As , the term dominates in the denominator if , so:
Thus, converges pointwise to 0 for all .
Step 2: Analyzing Uniform Convergence
To determine whether this convergence is uniform, we analyze the difference and find the supremum of this expression over .
We are given a hint to let and . Rewriting the function in terms of , we have:
[ g_n(x) = |f_n(x) - 0| = \left|\frac{x
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Uniform Convergence
Pointwise Convergence
Supremum
Formulas
f_n(x) = \frac{x}{1 + nx^2}
g_n(x) = |f_n(x) - 0|
Theorems
-
Suitable Grade Level
Undergraduate Level
Related Recommendation
Understanding Pointwise Convergence of f_n(x) = x / (1 + nx)
Pointwise Convergence of Sequence of Functions: f_n(x) = x/n
Uniform Convergence of $f_n(x) = \frac{\sin(nx)}{\sqrt{n}}$ over $\mathbb{R}$
Demonstrating Pointwise Convergence of the Function Sequence f_n(x) = x/n on [0,1]
Boundedness Theorem and Uniform Convergence Simplified Notes