Ramanujan summation: How does 1+2+3+4... = -1/12?

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Previous thread (with contributions from some of our mathematical Dopers who haven’t been around in awhile):

1+2+3+4+... = -1/12. In what sense?

I am currently reading a book called The Music of the Primes which is about prime numbers and in particular the Riemann Hypothesis. It discusses the work of the Indian mathematician Ramanujan and, almost in passing, mentions that his equation: 1+2+3+4+…=-[sup]1[/sup]/[sub]12[/sub] appeared at first to be the ramblings of a madman. It doesn’t really elaborate, so I can only assume that the equation is in fact cromulent. I looked on Wikipedia and found a page on Ramanujan summation, but that d…