A friend of mine just presented me with the following reasoning. Obviously, it's faulty since the result is absurd. Apparently, there's an assumption that's being made or broken against but I fail to see where.
S = 1–1+1–1+1–1+...
1 - S = 1 - (1–1+1–1+1–1+...)
1 - S = 1–1+1–1+1–1+1-...
1 - S = S
1 = 2S
Conclusion: S is 1/2. It means that the sum of (-1)^n would be a half. But that can't be true, can it? I would expect it to be either zero (each negative value cancels out precisely one positive one). Alternatively, I'd say that the result doesn't exist due to some black magic based on infinities and other voodoo stuff.
How can I point out the formal mistake made above?
The mistake is in the first step, where you assume $$S=1-1+1-1+1-1...$$
It simply assumes that the sum exists and is $S$, which is wrong, since we know that the sum doesn't exist.