Hey Veeky Forums, I still don't get how the sum of integrers can be -1/12, do you think you can explain it to me...

you know, if a mathematical theory defies common logic, it's probably the theory that's wrong and not the logic

It's not a theory; it's a proof. The dude in that Youtube video lays it all out using nothing but math you can learn in high school. It all stems from a logical combination of concepts that we know for a fact are true.

>Having more faith in your animalistic intuition than the universal truths of maths
I bet light can't be waves and particles at the same time either, right?

According to Euler 1 + 2 + 3 + 4 ... = INF.

I've seen the Ramanujan explanation, and it makes sense, but something is just wrong...

Under the normal definition of an infinite sum, this infinite sum diverges and thus has no finite value. When people "show that the sum is −1/12", what they really mean is something along the lines of "This sum is useful in many areas of physics, and if we are to assign any meaningful finite value to it, −1/12 is the only one that makes sense. Thus we may call the 'physics sum' −1/12." But because many people are not that uptight about abuse of notation, they decide to just call it the sum anyway, even though their definition of "sum" is different.

Further reading: en.wikipedia.org/wiki/Riemann_series_theorem

this was on the wikipedia article

>make up bullshit
>give it a variable
>math is always correct

I understand Ramanujan "summation" and consider it a mistake to regard the result as a "sum" in any way. I hold that the Ramanujan summation of the sequence [1, 2, 3, ...] is -1/12, but it doesn't have a sum.

Now what?

Oh for something like this it would have to be the king of all jooz

listen you idiot

the sum of all natural numbers is countable infinity.

That's just a fact.

a sum cannot equal more than one thing.

So the sum of all natural numbers cannot equal both infinity and -1/12

But we know it does equal infinity therefore it doesn't equal -1/12

we also know that positive numbers cannot be summed together to get a negative result.

this is true whether you do it finite times or infinite times. proof: by induction we know that adding more positive terms to a sum makes the sum further from being negative
we also know that an infinite sum necessarily has more terms than a finite sum
so we know that an infinite sum of positive terms necessarily takes its result further from being negative than a truncated, finite version of the same sum.

So no the sum of all natural numbers cannot possibly equal -1/12

that simply goes against basic axioms and properties of real numbers in any normal arena of maths (whether you're working in a vector field, working in R, working with linear algebra axioms , etc.)

Then how do explain the fact that this has been measured in nature via the Casimir force?

nigger what

You hear it cuck

It doesn't converge; neither using regular summation (epsilon definition), nor with Cesaro or Abel summation. Ramanujan summation is not equivalent to convergence of a normal sum.

Now I am curious.
For such a long post containing facts, you seem to forget that nothing can equal infinity because it is not a number and infinite sums don't equal numbers. The terms you were looking for are diverge and converge (latter being denoted by = as well does not make it mean "equal").

yes, but this does not really refute my argument since I could just reword it to:
the sum of all natural numbers is divergent and a sum that diverges cannot equal a finite number because if sum equalled a finite number then it would be a convergent series.

also the proof in the second part of the post remains

heh, please kid. the same way physicists thought that they had "measured" neutrinos travelling faster than the speed of light?
Thanks, but I'll take logical proof for how mathematics works over a "measurement"

Light is neither a wave nor a particle, but it has properties of both. Those just happen to be two things which we have knowledge of that we can reference it to. Good work proving his point though

A particle can only act like a wave by assuming a state whereby the Fourier transform is similar to itself.

The greatest triumph of mathematics has been and always will be the ability to call whatever bullshit your mind can concoct and call it 'x'. You're a fucking imbecile if you think this is not the case.

That's not what that guy or the OP asked.

The sum of all positive integers approaches infinity.

I watched this Youtube video proof
youtube.com/watch?v=w-I6XTVZXww

Video claims that: "the sum of all natural numbers is -1/12"

>Video: Let S_1 = 1 - 1 + 1 - 1 ...
>Video: "If there's odd number of elements in that sum, you get 1. If there's even number of elements in that sum, you get 0. So what should the answer be? We take the average, so S_1 = 1/2."

This is wrong; S_1 is not 1/2. It's 0.
S_1 = 1 - 1 + 1 - 1 ...
S_1 = n(1) + m(-1)
When n and m approach infinity, S_1 approaches zero. Therefore S_1 is zero.

>Video: Let S_2 = 1 - 2 + 3 - 4 ...
>Video: 2*S_2 = 1 + (-2+1) + (3-2) + (-4+1) ... = 1 - 1 + 1 - 1 = 1/2, so S_2 = 1/4

This is wrong too; S_2 = n(-2) + m(2)+1. When n and m approach infinity, S_2 approaches one. Therefore S_2 is one.

Then the video goes on trying to use the S_1 = 1/2 and S_2 = 1/4 to prove S = 1+2+3+4... = -1/12. But as S_1 is not 1/2 and S_2 is not 1/4, the proof is incorrect.

Very predictable results as any mathematician with brain will clearly see that 1+2+3+4... approaches infinity. But still, in the name of communication, we managed to disprove it analytically too, without resorting to intuition.

Your counterarguments, please.

its a meme.

The sum of the positive integers isn't [math]-\frac 1 {12}[/math], it diverges. If you look at the equation [math]\zeta(s) = \sum_{n=1}^ \infty \frac1 {n^s}[/math] where [math]s[/math] is a complex number, you see that it's undefined at [math]-1[/math], which would correspond to the sum of the natural numbers. However there's one particular way you can "extend" this function so that it's defined and differentiable everywhere, and this new function, called the analytic continuation, has a value of [math]-\frac 1 {12}[/math] at [math]-1[/math].

>analytic continuation
Zeta[s] is an entire function.

Wrong. Sum of natural numbers approaches infinity. All other answers and arguments are for retarded inbred dogs

Yeah you're right, zeta is the analytic continuation of [math]\sum_ {n=1} ^\infty \frac 1 {n^s}[/math]

Yeah but "zeta" or "continuation" or all other retarded mongrel shit you keep typing is not relevant to this issue. So stop talking about those and face the situation.

What's the situation?

"sum of natural numbers equals what number"

Then the answer is that it's not equal to any number, because it diverges.

so "infinity" then

Then explain the Casimir force?

physics is not used to prove things in mathematics

physics relies on measurements (and approximated-measurements...) which are used to construct elaborate statistics and probability mess riddled with handpicked constants

not to mention that all measurements are done by an observer who actively participates in the phenomenon being measured. when observer measures a thing, like casimir force, observer is actually just measuring changes in himself which physicists think somehow correspond to real world

Looks like some physics shit, what about it needs explaining?

A theoretical calculation of a force that involves assuming the sum of cubes = a negative number, that agrees with experiment

let S be the sum of the integers,
consider S - 2S +S :
1 + 2 + 3 + 4 + 5 ....
0 - 2 - 4 - 6 - 8 -....
0 + 0 + 1 + 2 + 3 + ...
=
1 + 0 + 0 + 0 + 0 + ....

Thus 0=1
Checkmate atheists.

This simple demonstration came from a coworker of Cedric Villani, -1/12 cucks BTFO!!

Here is a link (it's in frog) if someone wants to go deeper.
Basically a french science youtuber make the usual 1+2+3+...=-1/12 video and write a blog for the one wanting to see the maths, Remi Peyre (Cedric Villani coworker commenting under the nickname "Matheux") in the comment just BTFO the whole thing and go a bit deeper, showing that there are some divergent sum that can be attributed a value, but the sum of the integers is not one of them.

sciencetonnante.wordpress.com/2013/05/27/1234567-112/

divergent monotone sequence has a diverging series

also pic related is a clearer version.

yea so its evaluation is done by shifting the values of the second conlfated series what about the leftover terms that werent substracted? both r countable sequences so should have the "same" number of terms but we used a term in the second for not every one in the first

Could this kind of thing be extended further to other divergent series? It'd expand upon Riemann series theorem.

Riemann series theorem ? The one on conditionally convergent series ? What does this has to do with divergent series ?

because we're rearranging terms on a divergent series to get different results

No I'm not rearranging anything, they are all in order.

STOP

calling it a sum then you fucking spergs

AaCZSDVD

It isn't a sum in the usual sense. Stuff like this can be found via analytic continuation of a series. A series may be defined, f(x), in the domain [a, b] but is undefined outside of that. Via analytic continuation, a new function may be defined, g(x), that is equal to f(x) in the domain of [a, b] but also yields defined values outside of that domain. This is the case for the sum of integers.

For example, 1-2+3-4+... is a divergent series but via analytic continuation, it's possible to arrive at the value of 1/4 via the function 1/(1+x)^2.

>Your counterarguments, please.
Ramanjuan summation. You're allowed to manipulate sums to give unintuitive answers when you use the right principles. Take more math classes.

A... series?

>doesnt realize that the answers are derived from following the paths along surfaces of the reinmann zeta function, and that both infinity and -1/12 are valid separately defined results

Holy fuck THIS

Imagination has always been the core of mathematics. Math represents a certain inherent truth precisely because it is "made up" within the minds of mathematicians.

Its quite simple really.
Consider the first three terms of the sum 1 + 2 + 3, now 1 + 2 + 3 + 4 is more than 1 + 2 + 3 so clearly three terms is less than four. Following same logic consider the sum up until 1 + 2 + .. + (infinity - 1), this has to be less than 1 + 2 + .. + infinity which equals the actual sum. But then again infinity - 1 is infinity therefore the sum has to be smaller than itself therefore it equals -1/12.

this guy gets it.
we are just leaving the reals for an infintesimal half loop around the origin.
slide round to the back end of zero and voila we have the finite complex limit.

>complex analysis
>basic
Don't make such a stupid fucking thread again yo udumb mmotherfucker

Prove it, of course. You just mentioned it. I too can say something bullshit like Zikkelberg continuation proves that 1+1=3. There.

(I'm well aware who Ramanujan is btw)

-1/12 is not valid answer. Infinity is only valid answer.

Wrong

youtube.com/watch?v=sD0NjbwqlYw

The sum of all natural numbers is not equal to -1/12.

No one ever showed, proved or suggests that is even a thing. The Riemann Zeta function is specifically not defined by the same function for negative numbers as it is for positive ones.

Even mentioning "Riemann zeta function" when discussing the "sum of natural numbers" is logical error.

Nothing Riemann zeta function or any higher/complexer mathematical object does is relevant to summation of natural numbers you dumbass.

We have agreed on rules about infinite sums. It has some very counterintuitive notions at first, but they do all make perfect sense when you think more about it. For example the FACT that the cardinality of natural numbers and integers is SAME, BECAUSE we voted and agreed to define it as such.

So throw the Riemann zeta function into the garbage and use the right tools to solve the problem.