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Could someone jog my memory on this?

The order of operation between an $\int$ and $\sum_{n\in \mathbb{N}}$ is not always interchangable? Note that the sum is an INFINITE sum

Why is it that $\int \sum_{n \in \mathbb{N}} \neq \sum_{n \in \mathbb{N}} \int$

Is the reason because the integral itself is a sum and the order of "summing" actually matters? (I think it's Multivariable calculus related stuff now)

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    @JyrkiLahtonen: Okay …2012-07-06

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It is easier and more instructive to give a counterexample using sequences: For example $\lim_{n\to\infty}\int_0^1 n^2x^n(1-x)\,dx=1$ even though the integrand goes to zero everywhere in [0,1]. To understand what is going on here, note that the function in the integrand has a graph which is a tall, thin peak getting taller and taller and thinner and thinner as $n\to\infty$, while approaching $x=1$ from the left.

Taking differences, you can easily realize the sequence as partial sums of a series, thus providing the counterexample you seek. To be precise, consider $\lim_{n\to\infty}\int_0^1\sum_{k=0}^{n-1} \bigl((k+1)^2x^{k+1}(1-x)-k^2x^k(1-x)\bigr)\,dx,$ which is just a difficult way to write the limit above.

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    @jak: Sure, that is easy to arrange. For a trivial class of examples, consider cases where the integrals are all zero. You can also have each integral diverge, but the sum being integrable.2012-07-14
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Correct -- they cannot always be interchanged. For example $ \sum_{n=0}^\infty \int_0^{2\pi} \cos(t+n)\,dt = 0$ but $ \int_0^{2\pi} \left(\sum_{n=0}^\infty \cos(t+n)\right)\,dt $ doesn't even exist (the sum never converges).

However, if everything converges absolutely, that is, if either of $ \sum_n \int |f(n,t)| \,dt \quad\text{or}\quad \int \sum_n |f(n,t)| \,dt $ exists, then Fubini's theorem guarantees that the summation and the integral can be done in either order.

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    So I should always check if the summand converges first before attempting to switch the integration and sum operation?2012-07-07
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Switching an limit and an infinite sum constitutes the interchange of limit processes. Said interchanges often yield unexpected results.