Interchange of integrals and limits

In summary, an example of a convergent sequence of real valued functions of a real variable is given where the integral of the limit is zero and the limit of the integrals is 1. This is achieved by using a sequence where convergence is not uniform and taking the Lebesgue integral on the sequence would also result in the same outcome.
  • #1
symbol0
77
0
Can anybody please show me an example of a convergent sequence of real valued functions of a real variable where the integral of the limit of the sequence is different than the limit of the integrals of functions in the sequence ?

Thank you
 
Physics news on Phys.org
  • #2
Try a sequence where convergence is not uniform
 
  • #3
[tex]\lim _{n\rightarrow \infty } \left( \int _{0}^{1}\!{x}^{n} \left( 1-x
\right) \left( n+1 \right) \left( n+2 \right) {dx} \right) [/tex]
 
  • #4
So the integral of the limit is zero and the limit of the integrals is 1.
Thank you g_edgar
 
  • #5
If we were taking the Lebesgue integral on this sequence, we would also have the same result, right?
 

1. What is the interchange of integrals and limits?

The interchange of integrals and limits is a mathematical concept that allows for the exchange of the order of integration and limit operations in certain cases. It is used to simplify the evaluation of integrals involving variables that are dependent on a parameter.

2. Under what conditions can the interchange of integrals and limits be applied?

The interchange of integrals and limits can be applied when the integrand is continuous and the region of integration is a finite interval or a bounded region in the Cartesian plane.

3. How does the interchange of integrals and limits affect the value of the integral?

The interchange of integrals and limits does not affect the value of the integral if the conditions for its application are met. However, if the conditions are not met, the value of the integral may change.

4. Can the interchange of integrals and limits be applied to all integrals?

No, the interchange of integrals and limits can only be applied to certain integrals that meet the specified conditions. It is important to check the conditions before applying this concept to avoid any errors in the calculation.

5. What are some applications of the interchange of integrals and limits in real-world problems?

The interchange of integrals and limits is commonly used in physics, engineering, and economics to solve problems involving optimization, motion, and rates of change. It also has applications in probability and statistics.

Similar threads

Replies
11
Views
1K
Replies
31
Views
704
  • Calculus
Replies
9
Views
1K
  • Calculus
Replies
4
Views
1K
Replies
8
Views
265
  • Calculus
Replies
24
Views
3K
Replies
2
Views
136
Replies
3
Views
887
Replies
8
Views
147
Replies
11
Views
1K
Back
Top