Where's the error in this "proof"?

Question

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?

Look at summation of Grandi's series to see that this question has been the topic of a lot of research... — J.-E. Pin, Sep 24 '19 at 08:12

score 3 · Accepted Answer · answered Sep 24 '19 at 08:03

3

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.

answered Sep 24 '19 at 08:03

Martund

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Aha! That sneaky thing. Let me guess. If we pair each even and odd element, we get (infinitely many) pairs of positive and negative units cancelling each other out but the logic can't the extrapolated to infinitely many elements because there's no evenly many such elements (nor oddly many, since there's no last element to consider the parity of). Right? – Konrad Viltersten Sep 24 '19 at 14:44

Where's the error in this "proof"?

1 Answers1