I just asked a similar looking question, but this one is a different question. Suppose that $$\lim_{u \to \infty}\sum_{i=0}^{\infty}x_i(u)=L$$ where $x_i$ is a function of $u$. Then does it follow that $$\sum_{i=0}^{\infty}\lim_{u \to \infty}x_i(u)=L$$ whenever $\lim_{u \to \infty}x_i(u)$ exists for every $i$? If not, then what sufficient conditions would guarantee this equality?
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1It does not follow in general. But there are certain s ufficient conditions for which the interchange is legitimate. – Mark Viola Sep 22 '16 at 20:03
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1If you search this site a bit, you will probably find several questions which are rather similar. For example, Under what condition we can interchange order of a limit and a summation? and maybe also some other posts linked there. Checking the list of related question which is in the sidebar on the right might be a reasonable thing to do, too. (This is list generated by Stack Exchange software based on your question.) – Martin Sleziak Sep 23 '16 at 02:01
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The usual sufficient condition for $\lim_{a \to 0} \lim_{b \to 0} f(a,b) =\lim_{b \to 0} \lim_{a \to 0} f(a,b)$ is that $f$ is continuous at $(0,0)$. Here $f(a,b) = \sum_{i =0}^{\lfloor 1/b\rfloor} x_i(1/a)$ – reuns Sep 23 '16 at 02:06