Assume: (1+2+3+4...) = -1/12 and (1-1+1-1...) = 1/2
Separate into evens and odds (1+3+5+7...)+ (2+4+6+8)...
Sum of odds = (1+3+5+7....)... call it S(o)
add (1-1+1-1+1...). (1+1)+(3-1)+(5+1)+(7-1)... = S(o)+1/2
2+2+6+6+10+10... = S(o)+1/2
2(1+1 + 3+3 + 5+5...) = S(o)+1/2
2(2(1+3+5+7...)) = S(o) + 1/2
4 (S(o)) = S(o)+1/2
3(S(o)) = 1/2
S(o) = 1/6.
Sum of evens (2+4+6+8...) = 2(1+2+3+4...) = 2 (-1/12) = -1/6
Sum of all positive integers = Sum of odds+evens
(1+2+3+4...) = (-1/6) + (1/6)
-1/12 = 0
Other urls found in this thread:
Your Ph.D is in the mail.
If you move/rearrange terms in an infinite sum, you can make it equal to literally anything.
Lookup Riemann rearrangement theorem and consider suicide
>Assume: (1+2+3+4...) = -1/12 and (1-1+1-1...) = 1/2
Your first line is wrong. You can conclude anything from that.
>Assume: (1+2+3+4...) = -1/12 and (1-1+1-1...) = 1/2
Wtf is this garbage user
>Assume: (1+2+3+4...) = -1/12 and (1-1+1-1...) = 1/2
Kiero.
Yeah OP, you don't have to "assume" something that has already been objectively proven.
that's only true for sums that aren't absolutely convergent
Which the sums OP is talking about are.
I'm good at spotting mistakes.
I can't spot the mistake.
>mfw my world view is a lie.
Read up on Barnett's Identity, you stupid idiot.
The series doesn't Converge to a real number and it isn't absolutely convergent so you can't rearrange terms.
>implying to be smart
wow how revolutionary OP, how about this one?
1 - 1 + 1 - 1 +... =
(1-1) + (1-1) + ... =
0 + 0 + ... = 0
1 - 1 + 1 - 1 + ... =
1 + (-1+1) + (-1+1) + ... =
1 + 0 + 0 + ... = 1
0 = 1
> people still not understanding that numerically manipulating infinite sums is not the same thing as rigorous complex analysis
If you're gonna try to act smug at least do it in LaTeX so it looks better
>Assume: (1+2+3+4...) = -1/12 and (1-1+1-1...) = 1/2
absolutely anything can be deduced from a flawed assumption
>mental gymnastics; the post
>Assume: (1+2+3+4...) = -1/12 and (1-1+1-1...) = 1/2
The OP is showing that the commonly claimed "facts" of analytic continuation that the sum of all natural numbers = -1/12 and that 1+-1+1+-1... = 0.5 are wrong
that's the point.
he's proven them wrong by contradiction.
The sums in OP's post aren't even the same notion of summation than Riemann's rearrangement theorem is about.
Someone gets it.
>Veeky Forums user proves Ramanujan wrong
are you sure about that?
The assumptions are correct, they were proved a long time ago
>mathfag
>doesn't know Ramanujan
>kek
>Implying poo-in-loo calculus conartistry is better than rigorous Western & Arab proofs.
>Implying that Hardy, a western mathematician, didn't ship him to England because of his genius
>Implying Ramanujan has not been rigorously challanged.
>Poo-in-loo got imported by jet another traitorous non-rigorous calculus hack, who is being "praised" for applying a parabola to biology.
>Poo-in-loo makes assumptions directly contradicting the rigorous Riemann rearrangement theorem, like a flat-earther "proving" jack shit in an age of artificial satellites.
ftfy
What assumptions?
His first proof consists of substracting two divergent series, contradicting not only the Riemann rearrangement theorem, but just ignoring all counter examples. Retarded
>Hurr durr I am Ramanujan and infinity-infinity=this by somebody else derived fraction function=-1/12
His second, "rigorous" proof assumes stupid shit like a non-convergent is a number (therefore convergent), also ignoring all possible counterexamples and the theorem by the same guy who created some of the definitions that poo-in-loo used.
>Hurr durr I am Ramanujan and I define this non-convergent series as any superposition that suits my wishes so I can show again that my first outcome isn't false.
Physicists get a hard on from using these sort of flawed assumptions, but at least they give us back some quantum shit to enhance the human species.
Instead of just making statements, give the mathematical proof.
Should I also state the Riemann rearrangement theorem for you?
Go read the proof for yourself you fucking brainlet.
I am not your librarian or lecturer.
Obviously not retard, I want you to prove that the sum of 1+2+3+... is not = -1/12
Jesus Christ, you call Hardy and Ramanujan hacks and expect me to take you word for it?
>Hurr durr I need you to prove it for me
>Hurr durr Grandi's series is convergent!
Okay then brainlet, here is the proof:
Let [math]M\in\mathbb{Z}_{>0}[/math].
[math]\sum_{k=1}^{2M}k=\frac{2M(2M+1)}{2}>\frac{n}{2}=M[/math].
By the standard Cauchy definition it is divergent.
Obviously, it doesn't come anywhere near -1/12.
>Hurr durr, no no, that's not the proof I want. I want you to give me poo-in-loo's proof.
Like I said, go read his proof your fucking self and you will understand exactly what my previous post said.
Here: arxiv.org
Will you now also tell us that P=NP and want everyone else to prove that it's not? kys retard
>Hurr Durr Hardy and Ramanujan are big names so they must be geniuses!
Cauchy, Euler and Riemann are our rigorous gods, not those hacks who came later to destroy math with non-rigorous bullshit.
Fuck off.
Actually, Cauchy, Euler and Riemann are fucking retarded. Euclid is the TRUE god of mathematics. BTW, Riemanns rearrengement theorem only applies to conditionally convergent series. You stated yourself that the series is divergent, you contradicted yourself, moron.
BTW, if the math can describe physics (which it can) it's correct. Nobody gives a single fuck about your pure math faggotry
>Euclid is the TRUE god of mathematics
One of the true gods, and Cauchy, Euler and Riemann are one of the true gods, whether you like it or not. These are the people that revolutionized mathematics and in a rigorous way.
>BTW, Riemanns rearrengement theorem only applies to conditionally convergent series
>you contradicted yourself
You cancerous cunt, the theorem can be applied the moment you consider the difference of the two sums, which is, according to Pajeet poo-in-loo, a convergent sum.
Are so utterly stupid that you don't know shit about sums and keep discussing like you are smart or anything?
BTW, kys you fucking moron.
Rigorous math (and not muh physics meth which has not been proved to be correct but only fits as a theory) is what keeps Pajeet's and your bullshit contained in the toilet and away from true science.
...
>the absolute state of mathematics
Mathlets, when will they learn?
>1 - 1 + 1 - 1 + ... =
>1 + (-1+1) + (-1+1) + ... + -1=
>1 + 0 + 0 + ... + -1= 0
>0 = 0
>The OP is showing that the commonly claimed "facts" of analytic continuation that the sum of all natural numbers = -1/12 and that 1+-1+1+-1... = 0.5 are wrong
It is not a "fact" and it is not "commonly claimed" inside mathematics.
There is no doubt that in standard analysis these sums diverge.