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? 
 A: No physical experiment ever predicts the result of a mathematical formula. A physical experiment may determine whether a certain model, described in the language of math, applies to a particular physical phenomenon.
That being said, divergent series can come up when working within the mathematical framework of quantum field theory. The values of certain physical quantities, like scattering cross sections, can be expressed in terms of infinite series that diverge unless a mathematical technique called regularization, followed by renormalization, is used to allow the series to have finite sums. The value of the sum may vary depending on a value chosen for a parameter called a coupling constant used in the summation of the series. By comparing to experiments, the value of the coupling constant can be determined.
A: 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.  
