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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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