Quote:
Originally Posted by emerald000
Anyway, there is a branch of mathematics called analytic continuation. Without going into detail, it deals with how analytic functions behave outside of their domain. And there is this well-known function called the Riemann zeta function defined as the sum of 1/n^s for s from 1 to infinity. This function is only defined for s>1. But if we were to take s = -1, we would get 1 + 2 + 3 + .... And using those rules of analytic continuation, you can get a value outside of the range of functions and in this case, the value is -1/12.
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this.
it's like saying that just because you can get a maclaurin expansion of, for example:
(1-x)^-2 = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...
and the expansion is valid for |x|<1
so you could put x=0.1 and get 1 + 0.2 + 0.03 + 0.004 + ... = 100/81, which is valid
then you put x=10 and get 1 + 20 + 300 + 4000 + ... = 1/81
it's the same idea here, just replace "riemann zeta function" in whatever he said with "(1-x)^-2" and you'll see how ridiculous and nonsensical it is to talk of a divergent series converging to some value just because it has a non series representation outside where it converges
so there'd never be a situation where you'd have to add 1+2+3+4... directly and get -1/12, if you wanted to work out zeta(-1) you'd work out zeta(-1), not try to add the divergent series, which won't make any sense
edit: also non euclidean geometry sucks donkey balls, i did a course on that last year, and it was horrible and made me want to pull out all my hairs and superglue them to my chin. i think it was just badly taught, but maybe i just hate the subject