>autists will defend this nonsense

>>7093073
http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Zeta_function_regularization

kthxbai

>>7093073
Here's how your thought process should go unless you're an engineer.
> Hey, an equation.
> Wait, the left side is divergent.
> I wonder what the equality means.
> Ahh, it's analytic continuation.
> Cool, what can you do with it?

>>7093073
Apparently I'm not autistic enough then. Fuck this pseudo-math.

>>7093094
Zeta-function="Pseudo-math"?

>>7093107
> naming a function after the symbol used to describe it
good job autists

>>7093120
do you know your ABCs?

>>7093120
Not like it is a usual thing
http://en.m.wikipedia.org/wiki/Gamma_function

>>7093122
My alpha-beta-gamma?

>>7093107
Analytic continuation is not real math. It's handwavy physicist "math", i.e. "it works in experiment so it must be true even though it makes no sense".

>>7093126
Well how do you define real math?
Sure its implementation (in Sting theory if I am not mistaken) is based on your mentioned concept, yet the function itself is well-definend and proven

>>7093126
it is not handwavy. topology and any analysis course will teach you otherwise. do you know what an analytic continuation is?

>>7093682
don't get trolled, bro

>>7093126
But that's wrong.
>>7093129
It's used in all QFT, not just stringy things.

>>7093079
Thought process of an engineer
>What is this garbage
>Give me the answer, and if it's useful, I'll apply it to the real world.

>>7094068
Real thought process of an engineer
>This isn't a dick
>Give me more dicks to suck

>>7094068
I work with a lot of engineers and they really don't care for any math if they don't have to learn it for an exam.

>>7094089

probably because engineers care about applying their craft. Anything higher than Calculus III isn't even worth knowing in the real world. Hell, most jobs just require a decent knowledge of Calculus.

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>>7093126
>Analytic continuation is not real math
AHAHAHAHAAHAHHAHAHAHAHAHAH

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>>7094205

>>7093120
What would you propose calling it? A Riemann function?

>>7093073
Just solve for variable.
... = -85/12

>>7093073
this thread again. great.

>>7094068
Thought process of an engineer
>Can I suck it?

>>7093073
autists will make make a thread about it
autists will argue about it
people who didnt turn their brain off halway through school will facepalm and move on

>>7095214

Actually the middle dot is a multiplication sign, which means . . . is actually . squared, so the answer is the square root of -85/12

>>7093073

What do those dots mean?

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>>7095218

>>7094068
>Thought process of an engineer
>Give me the answer

Because god knows the engineer isn't going to figure it out on his own and needs the solution spoonfed to him like everything else in his pathetic "education."

>>7095645
That's assuming A=A

>>7095790
>if you sum up all POSITIVE NATURAL NUMBERS...
-1/12 is not the sum of all positive natural numbers, but of a series which contains them, and more.

>NO INFINITE SERIES IN NATURE
Infinity can not be grasped by language, which is why a word is used to describe its many concepts (imaginations) of "never ending."
Therefore there can't be infinite series in nature, as unfortunately as there can not be zeros in nature.
>It's a divergent series.
It's a series that resets every other term, but who never ends. 1,0,1,0. The sum (term 1 to term NOPE) can then only be described as 1 or 0. It does not increase nor does it converge.
The average of the two sums that are equally probable (although impossible) can be assumed to be the average (since even the idea of summing an infinity is an assumption)
(1+0)/2.

>This is when I stopped taking math seriously.
Good for you.
I assume you don't read books, talk to people, or read poetry either.

Know-nothing here. If they come to the unavoidable conclusion that all that stuff equals -1/12, why don't they just take it as proof that all or some their previous theories were false?

>>7095628
the one correct post

>>7095670
triple multiplication

>>7093073
And this kids, is why you never use the equality sign in the wrong context, because people just won't get you.

>>7093079
>what the equality means
this is where most people go wrong
the series isn't *actually* equal to -1/12, it is a magical equal sign

>>7093076
this is the best troll math i've ever seen

>>7095863
found the autist

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I stopped caring about math when I was introduced to the concept of analytic continuation. What a crock of shit. If your function can only be extended by inventing values that you didn't know, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

>>7093076
-3c would be -3-6-9-12...
this proof makes no sense

>>7094089
this is true but as an electro engineer i like to play with math in spare time since even those math things that are not applicable and required in real world are still there , defining our universe and things we cant touch but we know they are there

>>7096896
>analytic continuation
where is it even used? you sound very sure and i kind a believe this

string theory uses this factoid to validate the 24 spatial dimensions out of the 26 total dimensions in our universe.....it's a bullshit string theory thing, no mathematician gives a shit about it, any mathematician doesnt care about the infinite partial sum of a diverging sum

>>7096887
>wala

>>7096934
It's pasta, but was originally about imaginary numbers.

Engineeringfag here.

My understanding is that this isn't an arithmetic solution that you can perform algebraic operations on, rather an analytic one used to differentiate divergent series. For example, the sum of all natural numbers squared approaches infinity differently than the sum of all natural numbers, and this is represented by the two series giving different values using the analytic continuation method, which is more useful than just saying they both go to infinity.

Am I understanding it correctly?

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I've replaced the confusing symbol with a less ambiguous substitute. Please spread the word.

>>7096999
Now it makes sense. Thank you.

>>7096887
>>7096972
>>>/ck/

>>7096883
The series isn't equal to anything, it diverges.

>>7095863
Infinite series do exist in nature. Zeno's paradox is a rather rapidly converging infinite series.

There's nothing clever about lying and pretending that that equals sign means the same thing as normal equality in Rn.

call it "can , in a sense, be assigned to" or better yet "magical equals".

>>7095214
>>7095645
>-85/12
>not -121/12
>thinking 1 + 2 + 3 + 4 = 7

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>>7096991
Yes and no. Arithmetic doesn't allow to compute any infinite sum.
If you say
"Call \sum_{n=1}^\infty f(n) the number which comes closer and closer (with respect to the distance function on R) to the partial sums \sum_{n=1}^m f(n), as m goes towards infinity"
then this uses the limit and is hence also part of an analytic theory.
You're right in that it gives you more power to separate different expression from another.

-1/12 is the second Bernoulli number B2=1/6 times -1/2! and those often show up when you relate discrete quantities with their smooth interpolations.
For example, the linearization of the exponential function at 0 is "exp(0)·h" and the finite difference is "exp(0+h)-exp(0)". Now sure enough, you get the Taylor expansion

h/(exp(h)-1) = 1 - h/2 + h^2/12 + ...

More dramatically, there is this equation relating sums and integrals due to Gauss:

\int_a^b f(n) dn = \sum_{n=a}^{b-1} f(n)+(lim_{x\to b}-lim_{x\to a}) (1/2\,f(x)\,-\,1/12\,f'(x)+...)

For example, for a=2, b=4 and f(n)=n^2, you have

\int_2^4 n^2=(2^2+3^2)+(1/2)(4^2-2^2)-\frac{1}{12}2(4^1-2^1)

Now naively

\int_0^\infty f(n) dn = \sum_{n=0}^\infty f(n)-(1/2\,f(0)\,-\,1/12\,f'(0)+...)+lim_{x\to b}(1/2\,f(x)\,-\,1/12\,f'(x)+...)

In the standard theory, f(n)=n gives
"(undefined int) = (undefined sum) - (-1/12) + (undefined lim)".
Note that we speak of undefined expressions only in the meta-theory, limits are not functions.
The point is now that you get a consistent theory of infinite sums, in your sense, by dropping all other undefined terms. E.g. here
"(the sum over all n) = -1/12".
The theory isn't more or less inconsistent than the one that assigns 2 to
1+1/2+1/4+1/8+...,
but they are not compatible with one another.

>>7097075
You keep bringing that up, but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same.

>>7097108
>You keep bringing that up, but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same.
Saying 1 + 2 + 3 + 4 + ... = 1 + 1 + 1 + 1 + ... would also be incredibly misleading (Get it, the + is set union, and equality is in terms of cardinalities of sets, where each integer is identified with the set containing that integer)
We have conventions, and when your notation breaks the normal conventions, you should specify what it means if it differs from the normal conventions. OP equation is just as misleading as saying 1 + 1 = 0 without metnioning you are working in Z/2Z

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>>7097108
Maybe I should add the full formula:
http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula

And pic related is the -1/12 relation in Mathematica, when transferred to the standard theory of limits,
where you can't just cancel the undefined integral but instead subtract it form the sum.
After evaluation of the inner limits, it's this relation:

http://www.wolframalpha.com/input/?i=Limit[z+%28-1+%2B+z%29^-2+-+Log[z]^-2%2C+z+-%3E+1]

>>7096920
eurphoria

>>7093073
>autists will take this as a literal sum and get huffy

>>7097235
what is a literal sum?

>>7097271
my dumb layman way of saying this isn't dumb layman addition

>>7096999
thanks this really cleared things up for me

>>7097033
the equal sign begs to differ
of course, in actuality it is not an equal sign, it is a "fantastical equal sign"

>>7093079
Here's how your thought process should go unless you're an engineer.
> Hey, an equation.
> Wait, the left side is divergent.
> I wonder what the equality means.
> Ahh, it's analytic continuation.
> Cool, all this thinking makes me feel like sucking cocks

What's the fastest way for a physics student to learn about this "analytic continuation" so these damn threads will finally make sense?

I'm familiar with general principles of groups and sets and some topology. It has become very apparent that the -1/12 has nothing to do with addition like I learned in 1st grade.

>>7093073

What I love about this equation is what so many people seem to miss.

Its like the "Schrödinger's cat" thought experiement. The autistics and engineerers will argue about it.

They dont see the underlying deep thought behind it. ( Because they are autistics and Engineers).

What this equation so beautifully tells us is that the foundations of mathematics are totally fucked up. Yes, they work for shit like building bridges, sending probes to Mars, counting beans in a bean factory and other such mundane tasks. Just like Newtonian physics works for most common day applications. But no one would seriously suggest that Newtonian physics is superior compared to the Einstein's Relativity. Well, maybe Engineerers and Autistics would.

The crux of the matter is that modern mathematics do not describe reality. Nor is it a system of logical proofs. We have made some fundamental errors in our mathematical systems. Fundamental errors because this sort of bullshit would not happen in a perfect mathematical system. And no one knows what the errors are yet. And the guy who realises what the errors are and comes up with the new way of looking at mathematics will make Einstein look like an intellectual dwarf.

>>7097867
Or you know... it's a divergent series.

>>7097872
the fact the the series is divergent, i.e. no limit of it's partial sums exists, doesn't hold us from assigning -1/12 to it.
Of course, if, your notation is to denote the predicate D(s):=(the series S is divergent) by D(s):=(S = \infty), then you're bunt to confusion.

>>7097975
then why cant you math faggots accept evaluated at x=0 sinx/x EQUALS 0
why you gotta write lim and arrows and whatnot?

>doesn't hold us from assigning -1/12 to it.
it actually should
i have no problem with zeta(-1)=-1/12
but you cant write that the sum of all natural numbers EQUALS -1/12

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Look, it's very easy. You can't take the value at the end so you take the value at the beginning.

>>7097993
which means you cant use the notation
1+2+3+4+... because if you were to add them up, youd reach the END not the beginning, which is infinity
>oh but infinity is not a number
fuck you

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>>7093126
>2015
>not analytically continuing your functions

kill yourself

>>7097108
That you even bothered to typeset this bullshit is commendable.

>>7098014
You can, try plotting the smoothed asymptote of a convergent sum and look at it.

Just define the sum as the y-intercept.

>>7098045
>Just define the sum as the y-intercept
the most mathematician thing ever said
"just define it"

>>7094063
You got me curious, where can I read something about it(what particular topic involves it)?

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>>7098232
Keywords are regularization and renormalization.

In field theories you often set up theories with parameters which yet have to be measured, like the mass or the charge of a particle.
For example, you might conjecture that there is some mechanical quantity Y associated with an energy E=\frac{1}{2}m_Y(x'(t))^2. But that alone doesn't tell you what the value of m_X is. There might be an experiment, say measuring the inertia of Y, which tells you it. Say the experiment says m_Y should be about 7. If the predictions of the theories with m_Y=7 stand the test of further experiments, you might be on to something.

In the above case, we measured masses and adopted the values. A fundamental theory of elementary particles might actually predicts the masses (e.g. string theory, in principle), or at least correlates the masses/charges etc. strongly and that a priori (e.g. quantum field theories). Here the masses are determined in the way that the theory actually only works for particular values of it.

In field theories the energy terms are more complicated than just the velocity x'(t) squared, there are energy densities and whatnot.
Instead of doing actual renormalization theory here, consider the integral

E=\int_2^\infty \left(\frac{2}{x+2}+\frac{1}{(1+x)^2}\right)\,dx

The sign \int_2^\infty is defined as lim_{a\to\infty}\int_2^a.
The integral over 1/x to infinity is undefined, i.e. the integral is convergent.

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(cont.)

Say you have a physical situation for which no established theory works.
To explain what's going on, you might conjecture that maybe there is a yet undiscovered quantity Z, interacting with the system (historically: neutrinos, anti-particles) and you come up with a physical theory where the situation actually looks like

E=\int_2^\infty \left(\frac{2}{x+2}+\frac{1}{(1+x)^2}-\frac{1}{x}m_Z\right)\,dx

And now the integral has the chance to be convergent, but only if m_Z=2, in this case. Not that it matters here, but in that case the integral has the value 1/4-log(4).
You’re in the situation now that you can say:
>Guys, guys, I predict further experiments will hint at E being -log(4)+1/4=-1.136…!
That value is kinda random, so if this happens they will be even convinced that your Z-quantity theory with mass equal to 2 is correct.
The idea that there is a higgs is from the 70’s, i.e. 40 year before it was possible to make that particle scattering measurement. It was just conjectured to be there because it was the simplest solution to complete the energy expression of the standard model.

http://en.wikipedia.org/wiki/Renormalization#Renormalized_and_bare_quantities

Assining -1.136… to an integral over a positive integrant might seem completely arbitrary without any context, of course.
But as math isn’t arbitrary, there aren’t so many theories to modify self-consistent theories.
At least if you want to keep things simple. Ramanujan came up with a theory assigning -1/12 to 1+2+3+… somewhere in India after having read age old books.
Here is a computation in quantum electrodynamics using that result

http://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension

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(cont.)
The Ramanujan sum can also seen as a counter term situation.

1+2+3+4+…=lim_{z\to 1}(1z+2z^2+3z^3+4z^4…)=lim_{z\to 1}\sum_{k=0}^\infty kz^k

which for moderate z is

z\frac{d}{dz}\sum_{k=0}^\infty z^k=z\frac{d}{dz}\frac{1}{1-z}=\frac{z}{(1-z)^2}=\frac{1}{(1-z)^2}+\frac{1}{1-z}

We have

\sum_{k=0}^\infty z^k=\frac{1}{1-z}

and

\int_{k=0}^\infty z^k\,dk=-\frac{1}{\log(z)}

The smooth counter term for the divergent sum is

-z\frac{d}{dz}\int_{k=0}^\infty z^k\,dk=-\frac{1}{\log(z)^2},

which expands as

-\frac{1}{\log(z)^2}=-\frac{1}{(1-z)^2}-\frac{1}{1-z}-\frac{1}{12}-\frac{1}{240}(z-1)^2+…

>>7098408
isn't convergent*

>>7098427
z(d/dz)\sum_{k=0}^\infty z^k=z(d/dz)(1-z)^{-1}=z(1-z)^{-1}=(1-z)^{-2}+(1-z)^{-1}

-\frac{1}{\log(z)^2}=-(1-z)^{-2}-(1-z)^{-1}-\frac{1}{12}-(1/240)(z-1)^2+…

>>7097108
those equals are all more similar to each other than magical dur hur sum of negative numbers can make a positive number equals.

>>7098440
I'm not entirely sure what's your saying here, did you refer back to the post from yesterday on purpose?

>>7098463
I responded to
>but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same

>>7093073
>sum of positive integers is a negative fraction

when will people stop posting this meme