# Have experiments ever suggested two different values to the same divergent series?

I believe to have understood that some physical experiments suggest finite values to divergent series (please correct me if I'm wrong, my understanding of these matters is limited).

I heard, for example, that the "equality" $$\sum_{n=1}^{ \infty} n = - \frac{1}{12}$$ was suggested by some experiment conducted by physicists.

I was wondering if there are experiments in physics that seem to suggest two or more different values to the same divergent series. If not, why is this the case?

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That identity comes from the definition of the Riemann zeta function by analytic continuation - it's a purely mathematical procedure. I've never heard that it was suggested by a physical experiment. Any chance you can find a reference for that? –  David Z Jan 10 '12 at 22:21
Physical experiments don't give the results of mathematical equations, ever. They give the results of some physical process. Equations can either correctly predict the outcome of the physical experiment, or they can fail to do so. –  Colin K Jan 10 '12 at 22:24
@David Zaslavsky: Nope, I can't. I was told this was true by one of my mathematics teachers during a lecture he gave. –  Max Muller Jan 10 '12 at 22:28
This particular sum is also discussed here, and on math.SE here. –  Qmechanic Jan 11 '12 at 10:55

You might like to read the Wikipedia articles on 1+2+3+4+... and on zeta function regularization including the comments on heat kernel regularization, the evaluation of path integrals in curved spacetimes and the vacuum expectation value of the energy of a particle field in quantum field theory.

The origin seems to have been Leonhard Euler, who did do physics, but seems to have produced this particular assertion playing with mathematics. Srinivasa Ramanujan later produced it too.

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Is it regularization or renormalization? Im quite sure that the subtracting of infinities while dealing with the charge of an electron(Whicg is actually $\infty$) is called renormalization... –  Manishearth Feb 10 '12 at 8:20