1+2+3+4+5+..... = (-1/12)!

Yakuza

I saw this on my Facebook feed earlier today. Mind = blown! (I think some of the intermediate sums in the video below have featured here before, but I don't recall the overall result being demonstrated before)

Enjoy!

https://www.youtube.com/watch?v=w-I6XTVZXww

unkel

My daughters primary school maths book is called mathemagic

1 - 1 + 1 - 1 + 1.... = 1/2

LOL!

Cleverly done though, it has to be said.

I prefer the simplicity of this one:

S = 1 + 2 + 4 + 8 + .....
= 1 + 2(1 + 2 + 4 + 8 + ...
= 1 + 2S

=> S = -1

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

Davidius

Of course strictly speaking 1+2+3+... is normally understood to just be the lim N->inf of 1+2+...+N which diverges. Semantics and tomfoolery are afoot.

sponsoredwalk

Davidius wrote: »

Of course strictly speaking 1+2+3+... is normally understood to just be the lim N->inf of 1+2+...+N which diverges. Semantics and tomfoolery are afoot.

Divergent series show up in quantum mechanics, for example, and can be explicitly evaluated, when possible, to give a physically realizable meaning so surprisingly there's actually no tomfoolery about them if you're being careful. Cauchy's definition of convergence is just insufficient in some cases and a divergent series is sometimes just telling us that we're working with an inadequate and constrained representation of a larger function.

A nice analysis related to the video posted was given here a while ago.

Davidius

sponsoredwalk wrote: »

Divergent series show up in quantum mechanics, for example, and can be explicitly evaluated, when possible, to give a physically realizable meaning so surprisingly there's actually no tomfoolery about them if you're being careful. Cauchy's definition of convergence is just insufficient in some cases and a divergent series is sometimes just telling us that we're working with an inadequate and constrained representation of a larger function.

A nice analysis related to the video posted was given here a while ago.

I'm aware of the different means of assigning values to particular divergent series. However the issue is basically a semantic one while the video is geared toward ordinary people who likely have no analysis and won't be able to understand the difference between the limit of the sum and some regularisation/extension to a wider function. The tomfoolery comes through seemingly unjustified steps (doing ostensibly similar manipulations on other divergent series yields odd and inconsistent results). "Proof by physics" is not totally relevant to a purely mathematical result, reality is only a crude approximation after all.

Miggle Kop

The way I look at it is that an divergent infinite series can be considered equal to a finite number in that the series as a whole behaves like that number.

Think about the equality 1 + 2 + 4 + ... = -1. Now, -1 has the property that if we double it and add one we get back to that number. The series s = 1 + 2 + 4 + ... also behaves like this since 2s + 1 = 1 + 2 * (1 + 2 + 4 + ...) = s.

In that sense s = -1 even though it doesn't converge to -1.

Miggle Kop

Davidius wrote: »

"Proof by physics" is not totally relevant to a purely mathematical result, reality is only a crude approximation after all.

The fact that these kinds of sums work in physics surely shows that they have some kind of truth to them and aren't just algebraic tricks.

Davidius

Miggle Kop wrote: »

The fact that these kinds of sums work in physics surely shows that they have some kind of truth to them and aren't just algebraic tricks.

It shows that a regularised function of these sums has use in physical models (and physicists will argue this means the values there are a 'natural' choice for otherwise divergent series but I don't know how exactly string theory treats them or interprets them). It doesn't demonstrate that the kind of manipulations presented are useful in general for obtaining 'natural'/regularised values for a divergent series. "Flukes" or hidden non-general properties are always possibilities.

Miggle Kop

You think it's just a fluke they work in physics?

Davidius

Miggle Kop wrote: »

You think it's just a fluke they work in physics?

No I said that it can be a fluke when non-rigorous reasoning turns up a valid result. -1/12 can be a valid value to assign to the series 1/n^s at s=-1 when it is extended in some well-behaved fashion. That you can arrive at such a value in a particular case through these kinds of manipulations does not imply that you can do so generally. I'm talking purely about the line of reasoning and I doubt this is the actual one used in string theory, at least not without additional refinement.

equivariant

Miggle Kop wrote: »

The fact that these kinds of sums work in physics surely shows that they have some kind of truth to them and aren't just algebraic tricks.

The idea that physics somehow is closer to 'truth' than some 'algebraic tricks' is a strange one. Mathematics and therefore algebra (tricky or not) is surely a much better way of determining truth than physics

bnt

My interpretation is that this kind of thing happens when you muck around with infinity, which is where a divergent infinite series leads. Since n/0 = ∞ for any value of n, infinity is a kind of mathematical "black hole" where information is lost and anything goes.

I don't see how they can get that first answer, that S₁ = 1 - 1 + 1 - 1 + 1 - 1 + 1 ... = ½
It's defined as an infinite series, so (in my view) it is not valid to stop it at any point, odd or even - because then it would no longer be an infinite series. The result would not just be shorter, it would be a fundamentally different type of series (bounded, not infinite), so I don't accept that it's valid to say that either possibility (odd or even) gives a result, or that you can average the two invalid results you get.

Justin1982

Miggle Kop wrote: »

The fact that these kinds of sums work in physics surely shows that they have some kind of truth to them and aren't just algebraic tricks.

Any of the sum of divergent series that I've come across in physics should be taken with a pinch of salt. The one's I've seen appear to be open to interpretation or suggest that physicists need to find a better theory.

Take the adding and subtracting of infinities or divergent series in calculations in quantum electrodynamics for example. Before I studied the maths I was led to believe that the theory involved the magical adding and subtracting of infinities that ended up cancelling each other out. Most physicists that comment on this method are uneasy with this mathematical technique or feel infinity has to be taken seriously if it gives the right answers in the end that match the experimental results. But I think they take the Feynman diagrams too seriously.

Basically the Feynman diagrams represent all the possible outcomes for the history of a particular quantum interaction. Each diagram ends up having a certain calculated value which may end up being divergent. But the thing is that in the end certain different divergent diagrams end up cancelling each other out. So the physicists tend to calculate the diagrams first, realize they are infinite and then afterwards tend to discover that the same diagram or series of diagrams actually didnt need to be calculated as they actually matched off to cancel another diagram or series of diagrams. So the way I see it, nature or the elegent theory of QED is not really as stupid as the physicists. It seems to me to know to cancel the diagrams first and not waste its time calculating series of divergent diagrams. Or I like to think that nature is laughing at physicists using these Feynman diagrams which they don't need to.

I'm not explaining it brilliantly but I do think nature if smarter and more elegant than some of the stuff physicists produce that seems on first looking to have inherent divergent or infinite terms that seem to riddle the theory.

ZorbaTehZ

It also depends on where you are, for instance in the field of 2-adic numbers the expression 1+2+4+8+...=-1 is entirely correct.