Why is it that mathematicians love abstractions and generalizations...

it's not the -1/12 thing itself that triggers me but the fact that it's always some brainlet who doesn't understand anything about the result at all and think themselves "le kool mathematician" spouting their nonsense.

convergent is relative

yes, it's not convergent in analysis, i.e. with respect to the |·| norm on the reals

and the -1/12 is by no means only found in the context of complex analysis and the Riemann zeta function. there are many theories of sums where 1+2+3+4+.. equals this rational number

>There's nothing wrong with calling it the sum, just as there's nothing wrong with calling i a number. Sure, you've expanded the notion of what it means to be a sum (in a way that is consistent with the previous definition, i.e. yields the same answer where defined)
>[math]-\frac1{12}\,=\,\infty[/math]
Fucking kill yourself, redditor.

How I summarize the -1/12 = 1+2+3+4+... idiocy:

N*(N+1)/2 = -1/12
N^2 + N = -1/6
N^2 + N + 1/6 = 0

N = -1/2 +/- sqrt(1/3)/2

Both roots are negative and irrational, therefore there can be no Natural number for which 1+2+3+...+N = -1/12

...

never thought I'd spot such a retard on Veeky Forums
you don't even understand how they got that -1/12 do you?

Yes I do, I'm just pointing out how retarded it is. It means nothing, it's just a number gotten out of applying some equations wrongly.

lmao that's NOT what -1/12 means, kill yourself fucking brainlet

Go away, pleb

I see lots of anger in you, I suggest either some meditation or a black dick, whichever makes you feel better.

because idiots like you think it converges towards that in the usual sense

>therefore there can be no Natural number for which 1+2+3+...+N = -1/12
This is entirely true and also entirely besides the point

Indeed, to state a stronger theorem, in analysis, there is no finite or infinite sum of positive numbers that converges to a negative number.

However, this isn't an argument against the statement 1+2+3+...+ = -1/12

>However, this isn't an argument against the statement 1+2+3+...+ = -1/12
Then use another notation.

"no, you"

But yes. As edgy as I am in my head, I actually agree.

Then again, half the statements in Fourier analysis speak of function convergence [math] \lim_{n\to \infty} f_n(x) = f(x) [/math] and actually the theorem involves some norm like [math] \lim_{n\to \infty} \int |f_n(x)-f(x)| dx = 0 [/math]. Would be cumbersome to point out the literally 500 notions of equality.

Then again (again), we can keep on sucking the engineers cock and use a symbol for e.g. the Ramanujan sum just because it's not much used "in practice".

From a syntactical point of view, it makes more sense to say that The zeta function of -1 Equals -1/12, whatever the fuck that means. 1+2+3+...=-1/12 is "please put me in mental house because I have developed schizophrenia" tier

You don't only find the result in relation to the zeta function. You have

[math] \int _m^n f(x)~{\rm d}x=\sum _{i=m}^n f(i)-\frac 1 2 \left( f(m)+f(n) \right) -\frac 1{12}\left( f'(n)-f'(m)\right) + \frac 1{720}\left( f'''(n)-f'''(m)\right) + \cdots. [/math]

due to Euler. And with that you find infinitely many decomposition of powers such as

[math] \frac{1}{3}(4^3-2^3)=(2^2+3^2)+\frac{1}{2}(4^2-2^2)-\frac{1}{12}\,2\,(4^1-2^1) [/math]

that all involve the infamous -1/12. And, in a related vain, in analysis you find

[math] \sum_{n=0}^\infty n \, z^n - \dfrac {1} { \log(z)^2} = - \dfrac{1}{12} + {\mathcal O}((z-1)) [/math]

(pic related for the two diverging analytical expressions and their difference (yellow))

And then e.g. Ramanujans theory of sums given essentially the same results, where however the log would be systematically dropped

where the limit z->1 of sum over n*z^n on the right hand side is the same as 1+2+3+4+ in analysis (both not being a real number)

>undergrads and engineers
>analysis
>platonic

Wow! highschooler!

I don't think you understand. Zeta-function regularization is *not* the only approach which yields -1/12. See math.stackexchange.com/questions/1327812/limit-approach-to-finding-1234-ldots for instance.

No mathematician thinks that, brainlets/normies who have been shown the result in an incomplete context do

We know how inner products work otherwise basic QM doesn't work, that guy is just a moron

because people fail to understand under which circumstances the riemann function reaches - 1/12.. that is the problem