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A specific example:

The infinite series 1+2+3+4+5+.... occurs in a calculation for the Casimir Effect. This series is the Zeta Function at s = -1. Z(s) = 1/1^s + 1/2^s + 1/3^s + .... The analytic continuation of Z(-1) is -1/12 Is it correct to say that 1+2+3+4+5+.... = -1/12 ? If you use the value -1/12 in your calculation of the Casimir force acting on parallel conducting plates and then measure this force in the lab, will your calculation agree with experiment?

Another example:

The Zeta function at s = 0 yields the series 1+1+1+1+1+.... The analytic continuation of Z(0) is -1/2 This series occurs in String Theory when calculating the number of space-time dimensions. Is it correct to say that 1+1+1+1+1+.... = -1/2 and use this value in your calculation? The result comes out to be 10 dimensions. Should we expect nature to agree with this calculation?