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