Let $\{a_{i,k}\}$ a enumerable family of non-negatives real numbers, suppose that for all $i$, $\displaystyle\lim_{k\to\infty} a_{k,i}$ exist. There are conditions for $\displaystyle\lim_{k\to\infty} \sum_{i = 1}^{\infty} a_{k,i} = \sum_{i = 1}^{\infty}\lim_{k\to\infty} a_{k,i}$ holds?
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I think you question is already answered in here – johny Feb 14 '17 at 00:45
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This is related to Monotone and Dominated Convergence Theorem, and summation can be generalised to integrals of arbitrary measures. – Henricus V. Feb 14 '17 at 01:05