Why is it not more likely that mathematicians need to reexamine their previous assumptions and proofs than that thus...

Why is it not more likely that mathematicians need to reexamine their previous assumptions and proofs than that thus obviously false statement is true?

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en.wikipedia.org/wiki/File:Sum1234Summary.svg
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It is a better option

You're free to try to reinvent the wheel tho, no one is stopping you. Oh wait, it's fucking hard? That's the way the rest of them feel like.

Because it might be true even if it doesn't make any goddamn sense yet.

Literally the best country in North America

Wrong thread lmao

No. It's not true. If I add one rock to a pile, then 2, then 3 is my pile getting bigger or smaller? When does my pile magically dissappear in negative rocks?

This is nothing more than proof we should have taken mathematicians lunch money at some point

the statement is not "if you add up all the positive integers you get -1/12".
that's fucking stupid. you can't sum infinitely many things.
if you think that anyone with a proper mathematics education believes the above statement then you are delusional.

the real statement is that "there exists a method of assigning values to infinite series, which (a) agrees with the familiar limit-of-partial-sums when both are defined, (b) has great number theoretical significance, and (c) assigns the value -1/12 to the series whose summands are the positive integers."

please stop pretending that your deeply flawed understanding of pop math topics is accurate, it makes you look like a moron.
maybe read a fucking book. here's one: "introduction to analytic number theory" by apostol. it's big and yellow and has fun little squares on it you can look at when you're done pretending you can do big boy math.

It's an abuse of notation. The notion of "infinite sum" here is different then the usual.

Apparently, the proof makes sense under weird other methods of counting. I haven't bothered to look into that.

Obviously, the proof isn't true for the real world, but it doesn't matter, sense you can't have infinite objects in the real world anyways.

its more of a weird quirk of infinite sums than anything

You have a geometric intuition/model for why [math] 1/2 + 1/4 + 1/8 + ... = 1 [/math] should be true and in analysis you learn a theory taylored to make the geometric intuition "true". And then you go on and require all notions of infinite sum follow some intuition of yours.

Just because something is true for the standard math of finite objects doesn't imply it's true for an infinite setting. This is a totally different context.

Consider the claim
>if a set B is bigger than some smaller set S (i.e. |B|>|S|), then the set of all subsets of B, i.e. the power set PB, is bigger than the PS, i.e. |B|>|S| implies |PB|>|PS|).
This is evident for finite sets but unprovable in general (like the continuum hypothesis)

You can write down a theory of infinite sums where the sum of all natural numbers can't be evaluated, and you can write down theories where it's -1/12. And the latter is a value that various seemingly independent theories agree upon. Besides, the theories where the sum equals -1/12 are none that many people learn about (e.g. with sums alla Ramanujan or defined via analytic continuation).

That language is a little sloppy.
Even if you e.g. don't believe in an infinite amount of objects in your space, in case you believe in the contiuum of space, and that i.e. e.g. makes sense to consider an unending sequence of parting e.g. 1 inch into two parts, you'll end up with the need for mathematical theories that involve unendinging aspects.
And besides, the -1/12 results is used in quantum electrodynamics, the theory well suited to describe lasers and electronics on a microscopic level.

Huh, TIL

Also isn't space not continuous due to Planck length shit, or is the Planck length just a limit to observation

[math] \displaystyle
\zeta \neq \Sigma
[/math]

It's easy to confuse yourself with this shit but it's quite simple.

All that the ramanujan summation stuff, cutoff and zeta regularization does, is look at the smoothed curve at x = 0.
What sums usually do is look at the value as x->inf.

It's just a unique value you can assign to a sum, really they have many such values.

en.wikipedia.org/wiki/File:Sum1234Summary.svg

You're mistaken, the Planck length (basically a unit) doesn't affect the continuous model of space (or spacetime) that underlies essentially all quantum theories/quantum field theories. Planck length as limit of observation doesn't apply either.

Did you write a bot that always prints this post?

Why is not more likely that OP is a shitposting high schooler who's talking out of his ass because he hasn't even taken Calc 1?

The point of logic is to value consistency and rigor over intuition.

The fuck kind of equation is this?

Mathematicians don't claim that this is true. The method of obtaining this answer involved using a valid method of computing the sum of series, which is valid when you plug in numbers between -1 and 1. If you plug in -1, the method technically doesn't work because the series then diverges but if you forget about that fact, the sum of the natural numbers would be -1/12.

The sum of the natural numbers diverges but if you try to sum it anyway, you get -1/12.

>the real statement is that "there exists a method of assigning values to infinite series, which (a) agrees with the familiar limit-of-partial-sums when both are defined, (b) has great number theoretical significance, and (c) assigns the value -1/12 to the series whose summands are the positive integers."

You forgot the clause "(d) satisfies the familiar properties and axioms of addition and sums, as implied by the use of the summation sign".

You will no doubt notice that this clause is false.

>please stop pretending that your deeply flawed understanding of pop math topics is accurate, it makes you look like a moron.
Honestly, I feel this one is the fault of the mathematicians. Deeply flawed notation leads to deeply flawed understanding.

Because this sum exists in nature too. Ever heard of the Casimir effect?

I just think it's fucking cool that if you do analytical continuation of the rieman zeta function for 0 and 1 you get the same value as you would if you use partial sums of 1+1+1... and 1+2+3....

There aren't so many symbols, just because [math] \sum [/math] for finite sums has some properties, doesn't mean all uses must do so too.

>You forgot the clause "(d) satisfies ...".
>You will no doubt notice that this clause is false.
That doesn't make sense.
Yes, it doens't behave like finite sums. But that's no argument.

For example, if you add a finite quantity of rational numbers, you get a rational number. But using the theory of limits, i.e. standard infinite sums,

[math] 4 \sum_{k=0}^\infty\frac{(-1)^k}{2k+1} = \pi [/math]

and [math] \pi [/math] isn't rational.

You don't want sums of positive numbers to be a negative numbers. But again, that's just because you have a physical intuition for the sum defined in the context that's tailored for real geometry.

This makes no sense.

>i dont understand it it must be wrong

...

There should be a comma after that first it, brainlet.

The comma was omitted because I'm mocking them, using improper grammar to imply that they aren't smart. Also maybe see a proctologist for the stick up your ass.

>No comma after Also

Nice quads

You can never get to infinite rocks, so your point is not valid.

also the Lamb shift

physicsworld.com/cws/article/news/2016/dec/08/sonic-lamb-shift-detected-in-ultracold-atoms

Fucking this.

>look at the smoothed curve at x = 0
wrong

You need more points

The moment you start writing 1 + 2 + 3 + 4 + ..., your sum STOPS MAKING ANY GOODMAN SENSE.

You can twist it in any algebra you want, it will still be utterly senseless. Slap any algebra into it and you might find literally everything.

Case in point with 1 - 1 + 1 - 1 + 1 - ...
This sum diverges. The writing above (with the dots) has no meaning.

I can "show" it has 3 different values.
(1 - 1) + (1 - 1) + (1 - 1) + ... = 0

1 - [(1 - 1) + (1 - 1) + ...] = 1

s = 1 - 1 + 1 - ... = 1 - (1 - 1 + 1 - 1 + ...)
s = 1 - s
s = 1/2

See ? That's why 1 + 2 + 3 + 4 + 5 + ... doesn't work : the number literally doesn't exist.

The sum of 1 + 2 + 3 + 4 + ... , is undefined (many such cases). However, plotting a graph of sums that are defined makes a nice picture of lines that end abruptly where sums are not defined.

Recall the function, f : y = 2x over the interval 0

>You forgot
i said what i meant, you hubristic moron

you can't sum infinitely many things using the finitistic axioms of numbers. there are too many things to combine. as soon as you put an infinity on top of the sigma you toss your familiarity with anything out the window.

hence why i stressed that the shit you learn in calc is just one way of assigning values to the otherwise bogus symbology[eqn]\sum_{\text{whatever}}^\infty[/eqn]

x = 1 + 2 + 3 + 4 + ...
y = 1 + 1 + 1 + 1 + ... = infinity
x + y = 2 + 3 + 4 + 5 + ...
so x - (x + y) = 1
i.e. -y = 1
But y = infinity
So infinity = -1
shantih shantih shantih

IQ points that is

The real problem is that they used the "=" symbol. They should have invented a new symbol that means something else. Saying those two quantities are equal was just asking for ridicule.

Same damn thread every time.
We need a banner for these nincompoops.
IT SOMETIMES ACTS LIKE -1/12 BUT NOT ALWAYS
-1/12 AND ANY OTHER SEEMINGLY STRANGE ANSWER LIKE 6^0=1 OR 0.5! = sqrt (pi)/2 (or whatever it was) IS BECAUSE OF PATTERNS FOUND THROUGH REGRESSIONS.

This is a meme. Saying that the sum 1/2+1/4+1/8+ = 1 "deserves" the equal sign is bs. This result, while much more useful than OPs, is also just a theorem from SOME mathematical theory. Unless you're a mathematical platonist, there will be no reason to reserve this character for some particular theories.
E.g. in Peano arithmetic, 2+7=9. In modular arithmetic modulo 4, you have 2+7=1.
Both are theories helping you with some situations, and when you set up formal logic, you have some standard axioms for = and add your paritcular axioms, for =, on top of it. E.g. pic related 7 and 8, which are particular for nats, vs. the first few ones for =, such as a=b implies b=a, which is taken to hold for all theories that use equality.
Don't be a mathematical realist and then go on to choose your Platonism according to what you've found most useful.

No. If I have more things, the number is bigger. Always. There is no debate.

Step 0, a(x) = -0.8(3) = -1/12.

x = 1 + 2 + 3 + ... = infinity
y = 1 + 1 + 1 + ... = infinity
x + y = infinity
so x - (x + y) = infinity - infinity, which is undefined

How is this supposed to work, with two coefficients?

[math] f(x) = \frac{1}{12} \left( - 1 + a_1 \cdot x + a_2 \cdot x^2\right) [/math]

With two coefficients, you end up with

[math] f(x) = \frac{1}{12} \left( - 1 + \frac{15}{2} x + \frac{11}{2} x^2\right) [/math]

so that
[math] f(0) = -\frac{1}{12} [/math]

and

[math] f(1) = -\frac{1}{12} \left( 1 - \frac{26}{2} \right) = 1 [/math]

and

[math] f(2) = -\frac{1}{12} \left( -1 + \frac{37}{1} \right) = 1+2 [/math]

but then you don't have any more coefficients to fit anything into.

Dumbass, why don't you research why it's sometimes-1/12