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I'm hoping to find a lot of information for the case when we can say:

$$\lim_{a \to 1}{ \sum_{k=b}^c{f(a,k)} } = \sum_{k=b}^c{ \lim_{a \to 1}f(a,k) }$$

...or, more generally, when $a$ approaches some value from both sides.

So, essentially, when can we exchange limits and summations?

Martin Sleziak
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Matt Groff
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  • Uniform convergence. – IAmNoOne Sep 04 '14 at 16:57
  • @Nameless: Thanks - are there any other situations when the function is not uniformly convergent that we can switch the two? I'm just trying to get a general idea of all the possibilities... – Matt Groff Sep 04 '14 at 17:09
  • What is the range for k? Do you sum over k from some finite sets? Or over $k=1, \ldots, \infty$? – Martin Sleziak Sep 04 '14 at 17:47
  • These questions seem related: http://math.stackexchange.com/questions/123738/limit-summation-interchanging, http://math.stackexchange.com/questions/23057/under-what-condition-we-can-interchange-order-of-a-limit-and-a-summation – Martin Sleziak Sep 04 '14 at 17:48
  • @MartinSleziak: Oops - the range for $k$ is finite - I will correct the summation. – Matt Groff Sep 04 '14 at 17:49
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    For finite sums, the existence of $\lim_{a\to \alpha} f(a,k)$ for all $k$ is necessary and sufficient for the correctness of the interchange. – Daniel Fischer Sep 04 '14 at 17:53
  • I have removed ([tag:sequences-and-series]) tag. (I have included it when I thought that you are talking about infinite sum. For finite sums, ([tag:summation]) tag is appropriate.) – Martin Sleziak Sep 04 '14 at 17:55

1 Answers1

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If you have only finitely many summands, then the above is true if you read it this way: Assuming that all limits on the RHS exist, then the limit on the LHS exists, too, and this equiality holds. You can derive it by induction using the result for sum of two functions: Proving the limits of the sum of two functions is equal to the sum of the limits

If you are interesting also in a similar question for (countably) infinitely many summands, you can find some posts on this site which discuss this, for example, Limit Summation interchanging or Under what condition we can interchange order of a limit and a summation?

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Martin Sleziak
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