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Under what conditions is it true that an infinite summation of a sequence can be rewritten as a limit, i.e.

$$\displaystyle\lim_{n \rightarrow \infty} \sum_{j=1}^n a_j = \sum_{k=1}^\infty a_k$$

Or alternately, when can we rewrite the limit to infinity as an infinite summation?

EDIT: Taking user2388's example, consider $$\displaystyle \sum_{k=1}^\infty \frac{1}{k}$$ The sum does not exist. In such a case, would it still be valid to write:

$$\displaystyle\lim_{n \rightarrow \infty} \sum_{j=1}^n \frac{1}{j} = \sum_{k=1}^\infty \frac{1}{k} $$

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    $\begingroup$ Isnt that the definition ? $\endgroup$ – Rene Schipperus May 17 '14 at 1:07
  • $\begingroup$ @ReneSchipperus Do you mean to say that every time we see a term like the one on the left-hand side, we can replace it with the term on right-hand side automatically and vice-versa ? $\endgroup$ – curryage May 17 '14 at 1:24
  • $\begingroup$ @curryage Yes, the right-hand side of that equation is defined by the limit on the left-hand side. It is just short-hand notation for that limit. $\endgroup$ – bradhd May 17 '14 at 1:38
  • $\begingroup$ @Brad See my edit to the question. If either the summation or the limit does not exist, can we still write it as shown ? $\endgroup$ – curryage May 17 '14 at 1:41
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    $\begingroup$ Yes, I suppose so. Again, the right-hand side is just notation that refers to the left-hand side: that they are equal is not something that needs to be proved, but is a matter of definition. $\endgroup$ – bradhd May 17 '14 at 1:43
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Under the condition that such a limit exists. "Summation" can be formally written for just any sequence, even if it does not converge. However, writing something like $$lim_{n \rightarrow \infty} \sum_{j=1}^n a_j$$ implies that such limit exists.

If it converges then such a limit it by definition the sum of the series. Howe else could you define the result of such summation? Still, in this very notation you can write a series that does not converge, for example $$\sum_{k=1}^\infty \frac 1 k$$ This one has no sum (having taken enough its terms you can make it more than any given real number x) but the notation is valid.

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