Sum of all Positive Integers

This makes no sense.

So, if I was given a potion of immortality, and then offered to be given one dollar one day, two dollars next day, three next one and so on until eternity - I am being deceived by 1/12 dollars?

Its a pop-sci misinterpretation of an interesting result.

>Unique value
>They have many such values

Pic one.

No because you will never actually get to the end of all the natural numbers. All mathematics that involves infinity in any way is basically intellectual fraud. Including irrational "numbers".

Depending upon how you define convergence, infinite sums will have different values. You don't have to pic one. When considered under some convergence, C, a sum S will have unique value (C,S), You can consider S under different convergences (say A) to get different values, (A,S). That's the context in which 'many unique' values are ascribed to a sequence of numbers.

Best to keep simple on this site, or explain as you would to a child, statements that you make.

Let 1, 2, 3, 4, 5, ... be the group of integers which has infinite cardinality

It is very interesting question what 1+2+3+4+... is.

I'd figure it's some bigger infinity but not -1/12.

You've no idea what you're talking about. Also stop replying to yourself, it's pathetic.

For any a,b>0, you can prove in Peano arithmetic that a+b>0. However, an infinite sum is not a notion within that theory. Or general group theory, for that matter.
You can also prove that if you add two rational number a, b, that the result a+b will be a rational number. However, [math] \sum_{n=0}^\infty \frac {1}{n!} = e [/math].

In analysis, the sum 1+2+3+4+... can't be a finite value, because you can show that [math] \lim_{N \to \infty} \dfrac {1} {\sum_{n=1}^N n} = 0 [/math].
In analysis, the sum 1+2+3+4+... diverges. But analysis is a complicated theory (it involves sets and uncountable objects like the reals), not like the axioms of group theory or natural numbers give you.

What's the curve?

Are you fucking kidding me? Alright dipshit, Look at Cesaro convergence and its generalizations, absolute convergence, uniform convergence, pointwise convergence, Lp spaces and their convergences. I know what the fuck I'm talking about. Different convergences mean different things, even when they're defined over the same objects. Some convergences can be defined for multiple objects. Educate yourself. Goddamn I never bought into the brainlet meme but you sold me.

This is not a proof.

>I read somethings on Wikipedia

You seem mighty vex'd.

Phys: bullshit
Math: bullshit

Indians suck at math

Who wrote that? Holy shit, this is the kind of shit that you would see in a popmath video.

You know that what that guy is doing is implicitly reordering the sum in a convenient manner, right?

On the left hand side you have S - 2S + S which is meant to be interpreted as: sum all of 2, then sum all of -2S and then sum all of S. Which would be, find what S converges to, then add to it what -2S converges to, then add to it what S converges to.

But on the right hand side he instead is reordering and associating certain parts of S - 2s + S

So he reordered:

(1 + 2 + 3 + ... + n + ...) + (-2 - 4 - ... - 2n -2 - ...) + (1 + 2 +3 + ... + n + ...)

into

1 + (2 - 2) + ( 3 - 4 + 1) + ...

Which is also fallacious.

Basically, whoever thought this was a good counter argument is fucking retarded too. Which means, you are fucking retarded. Even more so than OP, who is just ironically shitposting. You are unironically shitposting because you thought this result was rigorous in any way. Consider suicide.

Brainlet try reading it again

If you're the same person I was responding to, I was vexed, but I'm done trying to convince you of something someone else stated. If you cannot force yourself to understand two different types of convergence, there's no point in continuing the conversation. Even had I looked up the concepts on wikipedia, the fact stands: under each of these ways of defining convergence of infinite sequences of objects there is a corresponding object that is unique up to that way of defining convergence. How I attained specific examples for you to explore yourself has no baring on my original statement's, and the other person's, validity. I hope you become better than you are.

I second the question where this is from. Moreover, as this shows that no summation technique with these 3 conditions could give -1/12, which condition(s) is the one to drop for zeta regularization and Ramanujan summation?

>Being this triggered by an internet shitposter.

Talk about insecurity.

>rerdering
Kill yourself brainlet.

autism