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/sci/ - Science & Math

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2807350 No.2807350 [Reply] [Original]

1+2+4+8+... = -1.

The series is divergent but equating the notions of "limit of partial sums" with "infinite sum" is a simplistic, arbitrary, and quite frankly rather fascistic imposition of "order" that stifles mathematical creativity. "Degenerate art", "deviant lifestyle", "divergent series" ... they're all expressions of the same mindset.

Also, 1+1+1+... = -1/2.

Discuss.

>> No.2807375

>equating the notions of "limit of partial sums" with "infinite sum" is a simplistic, arbitrary, and quite frankly rather fascistic imposition of "order" that stifles mathematical creativity.
So your solution is to work with no definition at all?

Cool story, bro.

>> No.2808346

>>2807375
An infinite sum is simply what you get when you take a well-defined infinite set of numbers and add them all together. Seems pretty clearly defined to me, with no mention of limits, convergence, etc. being required.

>> No.2808399

>>2808346
1 - 1 + 1 - 1 + 1 - ... = 0
1 + (-1 + 1) + (-1 + 1) + ... = 1

Same set of numbers, different answers. Not well defined.

>> No.2808497

>>2808399
That's true for Grandi's series, but it isn't always the case. For example, I've never seen a demonstration that 1+2+4+8+... adds up to anything other than -1.

>> No.2808630

bump

I should also mention that ...999.999... = 0.

>> No.2808637

>>2808630
Where's the "proof" for that? I can't find it anywhere lol...

>> No.2808655

>>2808497
I'm I the only one who is only seeing +s here?

>> No.2808657

>>2808346
>An infinite sum is simply what you get when you take a well-defined infinite set
What set theory are you working in? Zermelo-Fraenkel + Choice?
What's your definition of "number"? Is it the standard model of the real number system? (You know, the uniquely defined field which is a closed metric space that's also an archimedian ordered field?) If not, what is your definition of "number"?

>[...] of numbers and add them all together
How do you define addition of an infinite amount of "numbers"?
If you add them all together, will the result again be a "number"?

Waiting to hear from you.

>> No.2808661

>>2808655

i see it too, and its making all these people look fuckin retarded

>> No.2808664

also, 1+2+3+4+... = -1/12

>> No.2808676

>>2808637
...999.999... = ...999.0 + 0.999...
as we all know, 0.999... = 1
...999.0 = -1, since if you add 1 to it, you get ...000.0 = 0.
So ...999.999... = -1 + 1 = 0.

>> No.2808677

The Casimir Effect in physics uses

1 + 2³ + 3³ + ... = 1/120

The value is obtain through the analytic continuation of the Riemann zeta function. The physical justification is that under infinite quantum corrections the sum obtains that value.

The results aren't wrong, it's just on a different branch than you're used too, just like z^3 = 1 has three different complex answers, you'll usually use z = 1 but not always.

http://en.wikipedia.org/wiki/Casimir_effect

>> No.2808683

ITT:
>hey guise, I'll just say random stuff and not mention the definitions I'm using so I can troll you day and night and you can't disprove me

>> No.2808689

>>2808657
Sir, I work on intuition, like all great mathematicians.

>> No.2808694

>>2808655
Where is it written down that you can't add an infinite number of positive numbers together and arrive at a negative number? I know it seems absurd, but infinity regularly produces counterintuitive results.

>> No.2808696

i'm sorry for being a 'tard, but since when does infinity equal minus one?

>> No.2808695

>>2808677
>The value is obtain through the analytic continuation of the Riemann zeta function. The physical justification is that under infinite quantum corrections the sum obtains that value.
Please, do tell us more.

And no, the answer is not given in the article you linked.

>> No.2808697

>>2808689
WELL WHOOP DE FUCKING DO, EVERYBODY I GUESS IT'S TIME TO STOP WORKING WITH AXIOMS AND JUST DO THINGS BY INTUITION BECAUSE THAT'S HOW ALL GREAT MATHEMATICIANS WORK

>> No.2808705

>>2808689
>>2808694
>Sir, I work on intuition, like all great mathematicians.
Oh I see. I'll let you play alone then.

Btw, perhaps present your intuitive prove of the RH (I'm sure you got one) to the math society, maybe they'll give you the 1 million dollar price money. (Since you're obviously an accomplished and successful mathematician with your intuition.)

>> No.2808718

>>2808705
Not a fan of RH, as it happens. It uses analytic continuation, which is a copout as far as I'm concerned.

>> No.2808719

>>2808695
http://www.wolframalpha.com/input/?i=zeta%28-3%29

>> No.2808721

>>2808695

I learned this attempting to follow http://www-math.mit.edu/~KEDLAYA/18.785/calendar.html in high school, and as such my understanding is probably fundamentally flawed, you're better off asking a mathematician or a physicist.

If you are interested in the zeta functional equation (where the results itt derived from) see http://www-math.mit.edu/~KEDLAYA/18.785/funceq.pdf

>> No.2808739

>>2808721
I'm a math undergrad and I can tell you that what you're saying doesn't seem to be making too much sense.

You're probably telling it wrong, which you are destined to do anyway since you don't seem to understand it at any level.

I wonder why you feel entitled to bring it up in a math discussion at all.
Perhaps next time shut the fuck up when that urge to hurf all over the place tingles your retard-center in your brain.

Also a kind protip: that series you posted diverges. So either there's a fundamentally different notion of what this notation is supposed to represent (OP mentioned the standard definition already: limit of the partial sums for n tending to infinity) or you just copied/remembered it wrong.

>> No.2808776

>>2808695
>>2808739

<div class="math">\zeta (s)=\sum_{n=1}^{\infty }\frac{1}{n^s}</div>

http://www.wolframalpha.com/input/?i=zeta%28-3%29

>> No.2808782

>>2808739

Honestly I don't understand where this newfound hostility of yours came from. Perhaps you are mistaking me for someone else in this thread. However, I will now attempt to reply with an equal amount of hostility.

Analytic continuations are well founded within ZFC, the fact that you're a math undergrad is has no bearing on anything considering most of the math undergrads I have met have shown themselves to be complete retards.

Yes the principal branch of the sum diverges, congratulations, you have noticed that a sum which terms does not tend to zero diverges, amazing discovery, now prove that this means that there can be no analytical continuations. Be sure to relay this fact to the number theoretical community and the physicists, not only proving Riemann wrong in that the zeta function is analytically expandable to C, you're also automatically disproving RH and showing that the Casimir effect and all derivative forces are completely bullshit. Be sure to make keen emphasis on how you're an all powerful, all knowing math undergrad.

The sum 1 + 2³ + 3³ + ... may be interpreted as zeta(-3), which is equal to 1/120.

If you have any doubts as to the value of the sum please consult http://www-math.mit.edu/~KEDLAYA/18.785/funceq.pdf it will provide a step by step proof, I'm sure you as the all knowing math undergrad you are will be able to go through the proof there without needing someone to hold your hand.

>> No.2808789

>>2808782
>>2808721
>>2808739

Also since I am now hostile to you I forgot to laugh at how you probably failed complex analysis.

>> No.2808796

someone explain to a non-mathmatician how the fuck you can add positive numbers to get a negative number and how you can add integers to get 1/120.

>> No.2808800

>>2808796
Explain to me how you can take the square root of a negative number first.

>> No.2808809

>>2808796

http://en.wikipedia.org/wiki/Ramanujan_summation

>> No.2808810

>>2808796
Because of the curvature of the universe, all lines are actually giant 4-dimensional circles, including the number line.

>> No.2808812

>>2808782
zeta(-3) has nothing to do with the series you posted.

The series you posted simply diverges. What does it have to do with zeta(-3)? You obviously can't define zeta(-3) to be the series you posted, because that series fucking diverges.
(I feel like a fucking parrot but perhaps it'll come through to you that way.)

I'll repeat: you may be able to prove that zeta(-3) = 1/120 for an analytic continuation of the zeta function (which is initially defined as >>2808776
pointed out), but what THE FUCK has that to do with the series you posted?

>> No.2808813
File: 8 KB, 363x360, 1300625280777.png [View same] [iqdb] [saucenao] [google]
2808813

>>2808800
1) multiply with -1
2) extract square root
3) multiply with -1
4) ...?
5) profit!

>> No.2808820
File: 46 KB, 580x549, 1893math-jokes-motivational-get-real-pi.jpg [View same] [iqdb] [saucenao] [google]
2808820

>>2808800
>>2808813
Imagine:
-1 = i²

>> No.2808834

"I'll help" - says the number e.

>> No.2808846

>>2808812

> zeta(-3) has nothing to do with the series you posted.
Yes it does.

> what THE FUCK has that to do with the series you posted?
They are equal.

You're retarded, just go through the steps in http://www-math.mit.edu/~KEDLAYA/18.785/funceq.pdf and at the end insert 3 for s.

>> No.2808848

>>2808812
I can't tell if this is a troll or not. Do you... do you really not know what a negative exponent is?

>> No.2808851

>>2808782
>The sum 1 + 2³ + 3³ + ... may be interpreted as zeta(-3), which is equal to 1/120.
Oh I see. It can also be interpreted to be potatoes, you just need to make sure to mention it. Which you did not do. Now don't be all butthurt about it.

this here:
>>2808809
also is a fundamentally different meaning for the notation used.
Without saying what your notation is supposed to mean, you might as well not use it at all.
You're suffering from the same problem as OP; although OP is a blatant fucktarded troll and you obviously had some exposure to real mathematics.

>> No.2808860

ITT: high school failures discover summation methods on wikipedia and butthurt undergrads that have never heard of it throw a fit.

Just to make it clear: 1+2+3+4+... does not equal -1/12. Rather it is undefined. However terms like this one tend to creep up during quantization of certain physical systems (an easy example is the harmonic oscillator). To cope with such ill-defined expressions physicists came up with the idea of regularization. Don't as me why it works or how the came up with it (I'm a mathfag) but one widespread method is Riemann-zeta-regularization. In this one replaces 1+2+3+4+... by zeta(-1), where zeta denotes the unique analytic continuation of the Riemann zeta function to the whole complex plane. Turns out zeta(-1)=-1/12 and funnily enough when computing with this value the appropriate quantized energies actually can be predicted in experiments. Yes, our world is truly weird.

>> No.2808875

>>2808851

You're just angry because you don't know what the fuck I'm talking about despite you feeling like the fucking math boss around /sci/ because everyone around here are complete morons, face it, this is too advanced for you and your little childish mind to comprehend.

Take a couple of courses in analysis and come back, kid.

Be sure to mail Casimir on how he's completely wrong in interpreting 1 + 2³ + 3³ + ... as zeta(-3) (despite it just being the definition), I'm sure he and the physics community will mail you back admitting their mistake.

You're a fucking math undergrad, know your fucking place, retard.

>> No.2808880

>>2808846
>They are equal.
Nope. The definition of the series you posted is
<div class="math">\lim_{n \to \infty} \sum_{k=1}^n k^3</div>

inbefore
>BUT ANALYTIC CONTINUASHUN !!1

>> No.2808899

>>2808880

Now look at the definition of the zeta function, I'm sure even you will be able to notice some similarity.

>> No.2808906

>>2808880
<div class="math">\zeta (s)=\sum_{n=1}^{\infty }\frac{1}{n^s}</div>

<div class="math">\zeta (-3)=\sum_{n=1}^{\infty }\frac{1}{n^{-3}}=\sum_{n=1}^{\infty }n^3</div>

Is this really that hard for you?

>> No.2808917

>>2808906
omg, yes I am fucking aware of that.
Note though that (as a complex valued function) zeta(-3) is NOT DEFINED using the definition you posted.
That's what analytic continuation is about. You can't just equate it with the original definition.
God.

>>2808899
yeah, no, thanks for pointing that out. That entirely escaped me.

>> No.2808927

>>2808809
So it's not a real sum, just an abstract notation, kind of like how i isn't a real number but an abstraction which is useful

>> No.2808947

Hey kids, see http://en.wikipedia.org/wiki/Casimir_effect look at the calculation done by Casimir.

Problem, math undergrads?

>> No.2808976

> 2011
> people still getting trolled by zeta function regularization

>> No.2808985

>>2808927

Ramanujan summation isn't the same as zeta regularization, however they coincide because the values for the sums make sense.

>> No.2808992

I am more intrigued by dimensional regularisation, this is "only" analytic continuation, nothing really strange.

>> No.2809030

>>2808927
yup confirmed for equals not actually meaning "literally equivilent"

http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7

that doesn't sum to 1/4th. the geometric average converges to 1/4th. different things.

>> No.2809033

>>2808875

Fuck, that was brutal.

Not saging out of spite; just nothing to contribute.

But, damn was that brutal.

>> No.2809172

You know, I always thought dogs laid eggs.
And I learned something today.