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The non-converging sum 1 + 2 + 3 ... = -1/12 has applications in Bosonic String Theory, interestingly enough http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%...


The non-converging sum you have mentioned is also called Ramanujan Summation.


Not just that, it comes up in Casimir effect calculations.




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