When can interchange sum and integral

  • Context: Graduate 
  • Thread starter Thread starter Aziza
  • Start date Start date
  • Tags Tags
    Integral Sum
Click For Summary
SUMMARY

The interchange of sum and integral is valid due to the operations being independent rather than the variables. Specifically, the discussion highlights the application of the Dominated Convergence Theorem to justify this interchange when dealing with functions of n that do not depend on t. The key takeaway is that the legality of interchanging these operations relies on the properties of the functions involved and the theorems applicable to them.

PREREQUISITES
  • Understanding of the Dominated Convergence Theorem
  • Familiarity with integration and summation operations
  • Basic knowledge of functions and their dependencies
  • Concept of independent variables in mathematical analysis
NEXT STEPS
  • Study the Dominated Convergence Theorem in detail
  • Explore examples of interchanging sums and integrals in mathematical analysis
  • Learn about the Monotone Convergence Theorem and its applications
  • Investigate the implications of variable independence in calculus
USEFUL FOR

Mathematicians, students of calculus, and anyone involved in mathematical analysis who seeks to understand the conditions under which summation and integration can be interchanged.

Aziza
Messages
189
Reaction score
1
Why in the attached picture is it legal to interchange the sum and integral? Is it just because n is not dependent on t? note: (c1)n is just a function of n
 

Attachments

  • intgeral-sum.jpeg
    intgeral-sum.jpeg
    50.2 KB · Views: 746
Physics news on Phys.org
You would have to apply something like the monotone or dominated convergence theorem to prove that it is valid in this case.
 
The reason you can interchange is not because the variables are independent but because the operations are; that is, the integration and the summation.
 
Last edited:

Similar threads

Undergrad Original definition of Riemann Integral and Darboux Sums
  • · Replies 14 ·
Replies
14
Views
4K
Graduate Swapping Integrals and Sums: When is it Justifiable?
  • · Replies 9 ·
Replies
9
Views
11K
Undergrad On integral of simple function and representation
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad I would like opinions of the latest draft of my note - Integration
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Converting momentum sums to integrals in curved spacetime
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....
  • · Replies 6 ·
Replies
6
Views
3K
MHB The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Gamma function/partial fractions (a problem from my Insights article)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad In which sense(s) do square integrable functions go to zero at infinity?
  • · Replies 2 ·
Replies
2
Views
3K
Integration problem using inscribed rectangles
  • · Replies 14 ·
Replies
14
Views
2K