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Let $f:\mathbb{N_0}^2\to\mathbb{R}$. For what $f$ is it true that $$\lim_{n\to\infty}\sum_{k=0}^nf(n,k)=\sum_{k=0}^{\infty}\lim_{n\to\infty}f(n,k):=\lim_{m\to\infty}\sum_{k=0}^m\lim_{n\to\infty}f(n,k)$$ .

For example, if $f(n,k)=(1-k/n)^n$ satisfies this property, then $$\lim_{n\to\infty}(1/n)^n+(2/n)^n+...+(n/n)^n=\lim_{n\to\infty}\sum_{k=0}^{n}(1-k/n)^n=\sum_{k=0}^{\infty}e^{-k}=e/(e-1)$$

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If you look at this on discrete measure spaces, you could probably look at this question as an extension of Dominated Convergence, and then in one case I would say if $f$ is positive and ``integrable." – toypajme Sep 8 '13 at 19:30
this is answered by monotone convergence theorem for sequences. See the function $f$ should be increasing as $n$ increases. – Paramanand Singh Nov 9 '14 at 9:41

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