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?
When we can exchange the limits?
2
$\begingroup$
real-analysis
sequences-and-series
limits
-
0I think you question is already answered in [here](http://math.stackexchange.com/questions/23057/under-what-condition-we-can-interchange-order-of-a-limit-and-a-summation) – 2017-02-14
-
0This is related to Monotone and Dominated Convergence Theorem, and summation can be generalised to integrals of arbitrary measures. – 2017-02-14