Correct.
A sum has a limit, then and only then, when the respective series of numbers converges towards 0.
You may be confusing the series and the sequence. The set of numbers {1, -1, 1, -1, ...} is a sequence. The series is the sum of the sequence as in the OP. That particular series even has a name, Grandi's series, and it is divergent.
The sequence {1, 1/2, 1/4, 1/8, ...} converges to 0, but the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. However, there are sequences that converge to zero, but form divergent series such as {1, 1/2, 1/3, 1/4, 1/5, ...}.
I had a translation problem there. I thought series was the translation for "Folge", and sum for "Reihe".
Yes, I understand. Using the correct terminology now, a series cannot be convergent if the respective sequence does not converge to 0. I didn't mean to imply that this meant that all series of sequences that converge to 0 are also convergent, but merely that it is a necessary condition for a convergent series.
(Assuming I understand that correctly...with my limited undergraduate knowledge from math lectures in my economics program.
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