Previously, I'd been taught that convergent series "converge" to a certain number. Taking a convergent improper integral as an example, the value of the integral would be "definite" and not infinity.
However, every time I think about this, it doesn't make sense to me (I suspect it's probably because of a flaw in some basic understanding).
Anyway, my thinking is that since we are adding infinitely many values (as we do with convergent series), the sum of infinitely many values would be infinite. However, according to the accepted answer for this question,
[That is] incorrect. As long as the positive values an you are summing decrease to 0 fast enough, the sum... will be finite.
While I don't doubt this, it still doesn't give me any intuition. My thinking is still that, regardless of how fast you approach zero, since you have "infinite time" to sum "infinite numbers", your speed at which you approach zero wouldn't matter. The analogy doesn't really convince me. Could someone help clear up (conceptually) whatever misunderstanding I may be having?