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Disclaimer: I'm just a 7th grader who's smart, so don't trust me on this.
Alright, so there actually is a different proof that 1-1+1-1+1...=1/2. First, let's consider this way of solving this convergent series, namely 1+1/2+1/4+1/8.... We already know this sums to 2.
Let us call this series S. Then 2S=2+1+1/2+1/4+1/8... = 2+S.
We can use similar methods to work with divergent series and other series. If 1-1+1-1+1-1... equals S, then S+S=1-1+1-1+1...+1-1+1-1+1-1...
Shift the second S over by one term, and all terms after the first in the first S get canceled out, leaving us with 2S=1 and S=0.5.
Similar methods are used in the final proof.