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, ...}.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders half a beer. The third one orders a fourth of a beer. The bartender stops them, pours two beers and says, "You guys should know your limits."
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