When can interchange sum and integral

In summary, the interchange law for sums and integrals states that the order of integration and summation can be interchanged if a series of functions is uniformly convergent on a closed interval. This law can be applied to infinite series as long as they are uniformly convergent. The Weierstrass M-test can be used to prove the interchange law. However, there are exceptions to this law, such as when the series is not uniformly convergent or when the region of integration is unbounded. The interchange law has practical applications in simplifying the evaluation of integrals and is commonly used in fields such as mathematics, physics, and engineering.
  • #1
Aziza
190
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
 

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  • #2
You would have to apply something like the monotone or dominated convergence theorem to prove that it is valid in this case.
 
  • #3
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:

Related to When can interchange sum and integral

1. What is the interchange law for sums and integrals?

The interchange law for sums and integrals states that if a series of functions is uniformly convergent on a closed interval, then the order of integration and summation can be interchanged.

2. Can the interchange law be applied to infinite series?

Yes, the interchange law can be applied to infinite series as long as the series is uniformly convergent on a closed interval.

3. How can the interchange law be proved?

The interchange law can be proved using the Weierstrass M-test, which shows that a series of functions is uniformly convergent if the series of their absolute values is convergent.

4. Are there any exceptions to the interchange law?

Yes, there are some cases where the interchange law cannot be applied, such as when the series of functions is not uniformly convergent or when the region of integration is unbounded.

5. What are some practical applications of the interchange law?

The interchange law is commonly used in mathematical analysis to simplify the evaluation of integrals, especially in cases where the integrand involves a summation of functions. It is also used in various fields of physics, such as electromagnetism and quantum mechanics, to solve complex problems involving integrals and sums.

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