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1-1+1-1+1-...=1/2

In order to find the sum of this series, first of all, the sequence of the partial sums must converge. However, the sequence of partial sums does not converge. Rather, it oscillates between 0 and 1.
Thus, insofar as the sequence does not converge there is no reason to talk about a unique sum of the series. In this case we can prove anything we want: 1, 1/2...

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