Sequence of functions; uniform convergence and integrating

In summary, the conversation discusses the convergence of a sequence fn and the verification of a limit involving the sequence. The homework equations and attempt at a solution suggest finding the uniform limit of fn and calculating the integral to verify the convergence of the sequence.
  • #1
Appa
15
0

Homework Statement


The sequence fn: [-1,1] -> R, fn(x)= nxe-nx2 converges pointwise to f(x)= 0, x in [-1,1]. Can you verify the following:

limn->[tex]\infty[/tex] ([tex]\int[/tex][tex]^{1}_{0}[/tex]fn(x)dx) = [tex]\int[/tex][tex]^{1}_{0}[/tex] (limn->[tex]\infty[/tex] fn(x))dx

Homework Equations


Theorem: If fn is continuous on the interval D for every n and fn converges uniformly to f on D=[a,b], then

limn->[tex]\infty[/tex] ([tex]\int[/tex][tex]^{a}_{x}[/tex]fn(t)dt) = [tex]\int[/tex][tex]^{a}_{x}[/tex] (limn->[tex]\infty[/tex] fn(t))dt = [tex]\int[/tex][tex]^{a}_{x}[/tex]f(t)dt) for every x in D.

The Attempt at a Solution


The main idea is to find out whether the sequence is uniformly convergent on D= [0,1]. fn = nxe-nx2 is continuous for every n because et is always continuous. I tried something like this to verify uniform convergence:
fn(x) is uniformly convergent on D if there is a positive number [tex]\epsilon[/tex] >0 and an index N, so that
|fn(x) -f(x)|<[tex]\epsilon[/tex] for every n[tex]\geq[/tex]N and every x in [0,1]
[tex]\Leftrightarrow[/tex] |nxe-nx2-0|<[tex]\epsilon[/tex]
[tex]\Leftrightarrow[/tex] nxe-nx2 <[tex]\epsilon[/tex]

But I don't know how to go forward. How can I show that the greater the index n gets, the closer fn gets to 0? I tried comparison but couldn't find any sequence, the term of which would be greater than fn so that the sequence would still converge.
 
Physics news on Phys.org
  • #2
Finding a uniform limit for f_n isn't that hard. You just need to find the extrema of the functions as a function of n. Take the derivative, set it equal to zero etc. But I'm going to guess that what you are really expected to do is actually calculate the integral of f_n and verify that the limit of the integrals is zero.
 

Related to Sequence of functions; uniform convergence and integrating

1. What is uniform convergence?

Uniform convergence is a concept in mathematical analysis where a sequence of functions converges to a limit function at a uniform rate. This means that the difference between the value of the limit function and the value of each function in the sequence becomes smaller and smaller as the index of the sequence increases.

2. How is uniform convergence different from pointwise convergence?

Pointwise convergence only guarantees that for each point in the domain, the sequence of function values converges to the value of the limit function. However, uniform convergence guarantees that for every point in the domain, the difference between the function values and the limit function values becomes arbitrarily small as the index of the sequence increases.

3. What is the importance of uniform convergence in integration?

Uniform convergence plays a crucial role in integration because it allows us to exchange the limit of integration and the limit of a sequence of functions. This is known as the uniform convergence theorem and is a powerful tool for evaluating integrals of sequences of functions.

4. How do you determine if a sequence of functions is uniformly convergent?

To determine uniform convergence, we need to show that for any given epsilon (ε), there exists a natural number N such that for all x in the domain, the difference between the value of the limit function and the value of the sequence of functions is less than ε for all indices n greater than or equal to N.

5. Can a sequence of functions converge non-uniformly?

Yes, a sequence of functions can converge non-uniformly. In this case, the convergence may depend on the specific point in the domain and the difference between the function values and the limit function values may not become arbitrarily small as the index of the sequence increases. This is known as pointwise convergence.

Similar threads

  • Calculus and Beyond Homework Help
Replies
18
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
577
  • Topology and Analysis
Replies
2
Views
658
  • Calculus and Beyond Homework Help
Replies
26
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
499
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Back
Top