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We are given positive functions $f_n(t)$ with limits (as $t \rightarrow 0$) $\overline{f}_n$.

Consider the following statements

$f_n(t) \rightarrow \overline{f}_n$ as $t \rightarrow 0$ for all $n$.

$\sum_{n \geq 1} f_n(t) < \infty$ for all $t$ and $\sum_{n \geq 1} \overline{f}_n < \infty$

Do these imply

$\sum_{n\geq 1} f_n(t) \rightarrow \sum_{n\geq 1} \overline{f}_n$ as $t \rightarrow 0$?

If they do not what additional conditions are needed.

Sudheer
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    you are asking if you can switch the order of limits. See this http://math.stackexchange.com/questions/15240/when-can-you-switch-the-order-of-limits – Svetoslav Feb 20 '16 at 15:40

1 Answers1

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Let $f_n = 1_{({1 \over n+1},{1 \over n}]}$, $\bar{f}_n = 0$.

Then $\sum_{n \ge1} f_n = 1_{(0,1]}$, $\sum_{n \ge 1} \bar{f}_n =0$.

If you want a strictly positive counter example, consider $f_n+{1 \over 2^n}, \bar{f}_n+{1 \over 2^n}$.

copper.hat
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  • Thanks, it would be useful if I could find some conditions under which the convergence happens. – Sudheer Feb 23 '16 at 06:12