When can interchange sum and integral

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Interchanging the sum and integral is valid due to the properties of the operations involved, not merely the independence of the variables. The discussion highlights the necessity of applying the monotone or dominated convergence theorem to establish this interchange rigorously. The specific case mentioned involves a function of n that does not depend on t, which supports the interchangeability. The key takeaway is that the operations themselves allow for this interchange under certain conditions. Understanding these theorems is crucial for justifying such mathematical manipulations.
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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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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.
 
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