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:

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

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.