Math question: 1 + 2 + 3 + 4 + 5 + 6 +...

[Mark R]

So what does this equal?

I tried to look this up, but I think I broke my math.

Any sensible discussion, preferably without resorting to words like, Ramanujan Sum, Riemman Zeta and Dirichlet series would be welcomed.

Edit: Hmm. Forgot to add the pole.

Options:
1. +Infintity
2. +Infinity ^ 2
3. (1 ^ 100) ^ 100
4. Graham's number
5. 0.9999999....
6. -1/12
7. e ^ i pi + 1

 

[Atomic Playboy]

Any pattern with "..." at the end is assumed to go on forever, so the answer would be infinity. Are you looking for a formula to determine the value at any random point in the sequence, or are you just confirming that a never-ending formula is infinite?

 

[purbeast0]

op ... infinity is not a number, it's a concept of a number that never ends.

your #1 and #2 are the same thing, and anything with ... at the end means it goes on forever, which would mean the answer to your question is infinity.

 

[MomentsofSanity]

-1/12 of course.

 

[Dari]

n(n+1)/2

 

[dave_the_nerd]

http://en.wikipedia.org/wiki/Summation

(n*(n+1)/2)

 

[C1]

I think that is what the OP was looking for (ie, - 1/12)


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


Probably actual best answer.

1 + 2 + 3 + 4 + 5 + 6 +... ----> ∞


Ramanujan summation.
Should have made veeery clear that you can't just add a divergent series.

 

[MongGrel]

42 !

little joke.

 

[Bird222]

I think that is what the OP was looking for (ie, - 1/12)


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

This

 

[CLite]

I think that is what the OP was looking for (ie, - 1/12)


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


Probably actual best answer.

1 + 2 + 3 + 4 + 5 + 6 +... = ∞


Ramanujan summation.
Should have made veeery clear that you can't just add a divergent series.

Yeap.

3:08
"I'm going to shift it along a little bit"

I recommend he retake some of his math classes.

 

[DrPizza]

p -> q is true whenever p is false.

Hence the ability to demonstrate that the sum = -1/12 by using nonsense.

 

[Leros]

p -> q is true whenever p is false.

Hence the ability to demonstrate that the sum = -1/12 by using nonsense.

It's not nonsense. The result has practical meaning in physics.

 

[CLite]

It's not nonsense. The result has practical meaning in physics.

which is?

 

[videogames101]

yeah, analytic continuation - unless you're going to grad school for math/physics, you won't ever have a use for it - and it suffices to say that the answer is infinity, but if your curious look it up and you'll see why it's not useful for most of us,

and to clarify further after reading a bit, when people pull the comparison of the zeta function to the series of summations, they aren't exactly equivalent i'm fairly sure

but i'm not qualified to answer fully

 

[gotsmack]

Factorial?

so it's: n!

 

[uclaLabrat]

Factorial?

so it's: n!

Isn't n! a descending series, i.e. it has a finite endpoint?

I iz bad at teh maths

 

[DrPizza]

It's not nonsense. The result has practical meaning in physics.

1-1+1-1+1... is a *divergent* series. Its sum is not 1/2. Though, its Cesaro sum is 1/2, that is not a true sum.

 

[Paladin3]

This thread makes me want to go fishing. At least those little guys I can outsmart some of the time. Numbers? Not so much.

 

[dave_the_nerd]

Factorial?

so it's: n!

Factorials are multiplication, not addition.

Addition, it's called summation.

 

[DrPizza]

Also, from numberphile, a particular golden nugget:
http://www.youtube.com/watch?v=0Oazb7IWzbA&t=13m20s

Is assigning a value to the series really a sum? "I would say it's not exactly the sum, because the exact sum, as you see, it blows up, infinite."

 

[Dari]

1-1+1-1+1... is a *divergent* series. Its sum is not 1/2. Though, its Cesaro sum is 1/2, that is not a true sum.

Right, it's a ramanujan summation, not a convergent.

 

[Leros]

which is?

It's used in the computation for the Casimir effect (see http://en.wikipedia.org/wiki/Casimir_effect).

It's use is somewhat explained in the last part of this article http://plus.maths.org/content/infinity-or-just-112

 

[DaTT]

Divide it by zero...dare ya.

 

[Arcadio]

Any pattern with "..." at the end is assumed to go on forever, so the answer would be infinity.

Not true for convergent series.

 

[Born2bwire]

It's used in the computation for the Casimir effect (see http://en.wikipedia.org/wiki/Casimir_effect).

It's use is somewhat explained in the last part of this article http://plus.maths.org/content/infinity-or-just-112

Eh, that works because the Casimir energy is not the actual physical effect. The energy, at face value, is infinite as the photon modes that contribute to the energy are infinite. What is physical is the density of state of these modes with respect to the separation of the objects, which is taken as the spatial gradient (derivative) of the Casimir energy with respect to this separation. So while the energy is a divergent sum, its gradient is convergent because the higher order modes are of such small wavelength that they are not greatly affected by a small deviation in the scatterers (and thus the density of states remains unaffected as we go up in frequency). So we can renormalize the Casimir energy using any factor that we may wish as long as this factor drops out in the gradient. A typical factor is the energy when the objects are at infinite separation, which is what Casimir did in his original paper. In addition, he just windowed the frequencies by saying that beyond a certain frequency we have a cutoff function but this is unnecessary. With this treatment, you end up with an Euler-Maclaurin series that you can evaluate to a convergent sum. The zeta-regularization is another way of regularizing the divergent sum. Either way, it doesn't affect the actual physical result because the regularization drops out. Fortunately, you don't need to know a damn thing about zeta-regularization to be able to do a lot with Casimir theory. It actually is one of those nice quantum effects that exists on the macroscopic level so you can end up (through rigorous proofs) with a quasi-classical method of calculating the Casimir force.

This is an important part of Quantum Field Theory. In early Quantum Electrodynamics, one of the greatest problems they had were these divergencies and infinities. Regularization was an important breakthrough in removing these divergencies without affecting the final physics.

So yeah, they're doing some "don't look behind the curtain" hand waving here, but the subtle point is that the series 1+2+3+... is not the same series that we are truly evaluating when we say -1/12.