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This youtube video from Numberphile, http://youtu.be/w-I6XTVZXww

shows how the value is derived. In the video, one interviewee claims that "this result is used in many areas of physics". In the video, only string theory is mentioned.

Which areas of physics use or depend on the sum $$1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12?$$

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The sum $1+2+3+4+\ldots$ is not equal to $\frac{-1}{12}$. The series is divergent, and tends towards infinity, as the cameraman speculated near the start of the video.

However, $\frac{-1}{12}$ can be associated with the series $1+2+3+4+\ldots$, for example with analytic continuation. To quote the Wikipedia page on the subject:

Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

Evelyn Lamb (mathematics postdoc, University of Utah) wrote a blog article for Scientific American about this very video. Tony Padilla (the man explaining the "proof" in the video) writes about the series with more rigor on his own webpage. For more information, a question on math.SE would be appropriate.

As far as the physics goes, the Wikipedia page on the series goes into some detail, including bosonic string theory, the Goddard–Thorn theorem, and computing the Casimir force for a scalar field. However, I'm not familiar enough with the subject to expand on it. For more information, a question on physics.SE would be appropriate. (This answer was written while the question was still on skeptics.SE)

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