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Is there a theorem which says when we can interchange the limit and sum as follow:

$$\lim_{x\to \infty} \sum_{n=1}^{\infty}f(x,n)= \sum_{n=1}^{\infty}\lim_{x\to \infty}f(x,n)$$

Note: In my case the sum $\sum_{n=1}^{\infty}f(x,n)$ is finite at each finite $x\in \mathbb R$.

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    http://en.wikipedia.org/wiki/Dominated_convergence_theorem2012-03-23
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    http://en.wikipedia.org/wiki/Monotone_convergence_theorem2012-03-23
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    This is a general treatment of a connection between uniform convergence and the interchange of limits: http://math.la.asu.edu/~jss/courses/fall06/mat472/limit_interchange.pdf2012-03-23

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