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I was deriving the identity $$EX = \sum_{n=1}^{\infty} p(X\geq n).$$

I had to use an interchange of limit sums from $\sum_{n=1}^{\infty} \sum_{i=n}^{\infty} p(X=i)$ to $\sum_{i=1}^{\infty} \sum_{n=1}^{i} p(X=i)$ but I don't know how to justify this. It's obvious that the inner sum $\sum_{i=n}^{\infty} p(X=i)$ is bounded but why is $\sum_{n=1}^{\infty} \sum_{i=n}^{\infty} p(X=i) < \infty$ ? You need that in order to do the swap of summations so I would appreciate an explanation for why we can do the swapping of sums here.

Thanks!

Kashif
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    Orders of sums and integrals can be swapped if all terms are nonnegative, regardless of whether they diverge to $\infty$ or converge. That's called Tonelli's theorem. There's also Fubini's theorem, which says if the double sum or double integral of the ABSOLUTELY VALUES of the terms is finite, then the order of sums or integrals, WITHOUT the absolute value signs, can be swapped. Together these imply that the only cases in which the swap is not valid is when the sum or integral of the positive terms and also the one of the negative terms, are both infinite. – Michael Hardy Sep 23 '13 at 23:11

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