why 1+2+3+4+... is not -1/12

S = 1 + 2 + 3 + 4 + ...

2S = S+S = 1 + 2 + 3 + 4 + 5 + ...
         +         1 + 2 + 3 + ...

         = 1 + 2 + 4 + 6 + 8 ... = 1 + 2*(1 + 2 + 3 + 4 + ...)
         = 1 + 2S

=> 2S = 1 + 2S
If your manipulation is valid manipulation, then so is mine.

But clearly there is no solution with finite S.

For sak of contradiction, if we assume S is finite as you claim, then we can correctly subtract 2S from both sides giving 1 = 0, contradicting 2S being finite.

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Trying to prove this via the zeta function involves an error in complex analysis.

What a complex function is equal to, and how that function may be represented, are two different things. Sometimes a respresentation can be valid in one region, but invalid in another region.

The zeta function has a pole at z=1. And even though the zeta function is smooth at z=-1, the summation is invalid at that point.

This is the same kind of problem we get in the geometric series formula: S_k = Sum{k^n} for n=0->inf

S_k = 1 + k + k^2 + k^3 + ...
=> k*S_k = k + k^2 + k^3 + ... = -1 + S_k
=> -1 = (k-1)*S_k => S_k = 1/(1-k)

So there we have two ways of representing S_k, firstly as a summation, and secondly as the function 1/(1-k).
S_k is extended over the complex plane BY THE SECOND FUNCTION, and note that it has a pole at k=1.

While the second representation is smooth and finite at k=2, the pole makes the summation invalid about that region.

By the second representation, S_2 = 1/(1-2) = -1.
By the first representation, S_2 = 1 + 2 + 4 + 8 + 16 + ... which does not equal -1.


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The reason why 1+2+3+... = -1/12 appears to "work" in physics could be down to several things. But what seems most likely is this...

The physics theory actually uses the zeta function, although it may not be clear that this is the case. Sometimes it can be hard to realise what functions we're are really working with, especially if the foundation we are building upon is oversimplified and thus also lacking this insight.  So in the derivation of this theory we've naively ended up with this summation in there representing zeta. And -1/12 is the correct evaluation of the zeta function at z=-1, even though the summation is an invalid representation of the zeta function in the region of the complex plane.