# Do you agree 1 + 2 + 3 + 4 + 5 + ... = -1/12???

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#3

No, mathematically this is nonsense. All they've done is redefined what mathematicians have defined as convergence which, before anyone tries arguing, is a very basic definition in real analysis that underpins a significant amount of mathematics - and it just so happens that this re-defining is useful in certain applications.

Clearly, to anyone who is mathematically inclined, the idea of saying the series

Converges to is utter nonsense mathematically, there doesn't exist a such that for all , for so it doesn't converge by the standard definition of convergence.

I won't even get started on how violated mathematicians will feel at the thought of a sum of positive values somehow giving a negative result. Needless to say though, if you start redefining basic definitions in mathematics you're going to get results that don't make mathematical sense, whether they're helpful or not is another matter.

EDIT - TL;DR - They're playing around with sums that diverge, which is a mathematical equivalent of going full-******.

Clearly, to anyone who is mathematically inclined, the idea of saying the series

Converges to is utter nonsense mathematically, there doesn't exist a such that for all , for so it doesn't converge by the standard definition of convergence.

I won't even get started on how violated mathematicians will feel at the thought of a sum of positive values somehow giving a negative result. Needless to say though, if you start redefining basic definitions in mathematics you're going to get results that don't make mathematical sense, whether they're helpful or not is another matter.

EDIT - TL;DR - They're playing around with sums that diverge, which is a mathematical equivalent of going full-******.

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#4

No, I hate those stupid pattern/infinity videos on numberphile.

Ones like this are my favourite

Ones like this are my favourite

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#6

"Let's just take the average. The answer's 1/2." To translate Noble's post for anyone non-mathsy, that's what's wrong with this. No way is any term of that sequence ever going to be 1/2, not even the infinityth term. That's because it doesn't converge on 1/2, it stays the same distance away ad infinitum.

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#9

(Original post by

No, mathematically this is nonsense. All they've done, and by they I mean mostly physicists, is redefined what mathematicians have defined as convergence which, before anyone tries arguing, is a very basic definition in real analysis that underpins a significant amount of mathematics - and it just so happens that this re-defining is useful in certain applications.

Clearly, to anyone who is mathematically inclined, the idea of saying the series

Converges to is utter nonsense mathematically, there doesn't exist a such that for all , for so it doesn't converge by the standard definition of convergence.

I won't even get started on how violated mathematicians will feel at the thought of a sum of positive values somehow giving a negative result. Needless to say though, if you start redefining basic definitions in mathematics you're going to get results that don't make mathematical sense, whether they're helpful or not is another matter.

EDIT - TL;DR - They're playing around with sums that diverge, which is a mathematical equivalent of going full-******.

**Noble.**)No, mathematically this is nonsense. All they've done, and by they I mean mostly physicists, is redefined what mathematicians have defined as convergence which, before anyone tries arguing, is a very basic definition in real analysis that underpins a significant amount of mathematics - and it just so happens that this re-defining is useful in certain applications.

Clearly, to anyone who is mathematically inclined, the idea of saying the series

Converges to is utter nonsense mathematically, there doesn't exist a such that for all , for so it doesn't converge by the standard definition of convergence.

I won't even get started on how violated mathematicians will feel at the thought of a sum of positive values somehow giving a negative result. Needless to say though, if you start redefining basic definitions in mathematics you're going to get results that don't make mathematical sense, whether they're helpful or not is another matter.

EDIT - TL;DR - They're playing around with sums that diverge, which is a mathematical equivalent of going full-******.

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#10

(Original post by

I swear this makes perfect sense? If S=1-1+1-1+... then S=1-(1-1+1-1+...) which is the same as S=1-S, then 2S=1 and S=0.5?

**QuantumHatter**)I swear this makes perfect sense? If S=1-1+1-1+... then S=1-(1-1+1-1+...) which is the same as S=1-S, then 2S=1 and S=0.5?

It is a partial series of S, not S itself because the first 1 is not included.

The problem is the infinity here and that the series doesn't converge.

Because when S=1-S, and S converges to 1, it would mean 1 = 1 - 1 => 1 = 0 and this is wrong.

Or if S converges to 0, it would mean 0 = 1 - 0 = > 0 = 1.

So as you see it does not make any sense because it does not converge and you change the definition of S.

It is always messy when you divide through 0 or play with infinity in any kind of form.

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#13

(Original post by

The problem here is your definition of S changes. You say "S=1-1+1-1+... ". So everything from begin (first 1) belongs to S. But then you say "S=1-(1-1+1-1+...)" and define S starting from the second 1. It is no more S.

It is a partial series of S, not S itself because the first 1 is not included.

The problem is the infinity here and that the series doesn't converge.

Because when S=1-S, and S converges to 1, it would mean 1 = 1 - 1 => 1 = 0 and this is wrong.

Or if S converges to 0, it would mean 0 = 1 - 0 = > 0 = 1.

So as you see it does not make any sense because it does not converge and you change the definition of S.

It is always messy when you divide through 0 or play with infinity in any kind of form.

**sbj**)The problem here is your definition of S changes. You say "S=1-1+1-1+... ". So everything from begin (first 1) belongs to S. But then you say "S=1-(1-1+1-1+...)" and define S starting from the second 1. It is no more S.

It is a partial series of S, not S itself because the first 1 is not included.

The problem is the infinity here and that the series doesn't converge.

Because when S=1-S, and S converges to 1, it would mean 1 = 1 - 1 => 1 = 0 and this is wrong.

Or if S converges to 0, it would mean 0 = 1 - 0 = > 0 = 1.

So as you see it does not make any sense because it does not converge and you change the definition of S.

It is always messy when you divide through 0 or play with infinity in any kind of form.

I would agree they wouldn't be the same if they didn't go on to infinity, but they do, so S still equals 0.5 as far as I can see!

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#15

(Original post by

Infinity is indeed messy, but I do think that my definition does not change. I have lost the first one, but because there are an infinite number of terms,

I would agree they wouldn't be the same if they didn't go on to infinity, but they do, so S still equals 0.5 as far as I can see!

**QuantumHatter**)Infinity is indeed messy, but I do think that my definition does not change. I have lost the first one, but because there are an infinite number of terms,

**the sum is exactly the same**. Infinity minus one is still infinity is it not?I would agree they wouldn't be the same if they didn't go on to infinity, but they do, so S still equals 0.5 as far as I can see!

This works 0 = 0 or 5+6 = 11 but this does not work ∞ = ∞.

So is infinity minus one is not the same as inifinity.

Because it is infinite, it has no end. You can't make an equation, it is NOT equal.

My infinity is 10^50 and yours 10^49, is that equal? When do we have an equation, like the same? We never have.

As if "S1=1-1+1-1+... " isn't equal "S2=1-1+1-1+... ", so S1 ≠ S2.

Because you don't know where it ends, there is no result, but you have infinite sums.

I could also make "S=1-1+1-1+... " -> "S=1-1+(1-1+... " -> S = 1 - 1 + S -> 0 = 0 -> No information about S.

But it does not converge, it can't be 1 or 0 or 1/2.

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#16

(Original post by

I Infinity minus one is still infinity is it not?

**QuantumHatter**)I Infinity minus one is still infinity is it not?

∞ = (∞ + 3) | -∞

0 = 3

or

∞ = (∞ - 1) | -∞

0 = -1

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#17

(Original post by

The bolded part is the problem.

This works 0 = 0 or 5+6 = 11 but this does not work ∞ = ∞.

So is infinity minus one is not the same as inifinity.

Because it is infinite, it has no end. You can't make an equation, it is NOT equal.

My infinity is 10^50 and yours 10^49, is that equal? When do we have an equation, like the same? We never have.

As if "S1=1-1+1-1+... " isn't equal "S2=1-1+1-1+... ", so S1 ≠ S2.

Because you don't know where it ends, there is no result, but you have infinite sums.

I could also make "S=1-1+1-1+... " -> "S=1-1+(1-1+... " -> S = 1 - 1 + S -> 0 = 0 -> No information about S.

But it does not converge, it can't be 1 or 0 or 1/2.

**sbj**)The bolded part is the problem.

This works 0 = 0 or 5+6 = 11 but this does not work ∞ = ∞.

So is infinity minus one is not the same as inifinity.

Because it is infinite, it has no end. You can't make an equation, it is NOT equal.

My infinity is 10^50 and yours 10^49, is that equal? When do we have an equation, like the same? We never have.

As if "S1=1-1+1-1+... " isn't equal "S2=1-1+1-1+... ", so S1 ≠ S2.

Because you don't know where it ends, there is no result, but you have infinite sums.

I could also make "S=1-1+1-1+... " -> "S=1-1+(1-1+... " -> S = 1 - 1 + S -> 0 = 0 -> No information about S.

But it does not converge, it can't be 1 or 0 or 1/2.

I'm not very good at explaining things but I think this sum is similar to this question "show 1/9 =0.111111..."

Let x= 0.11111111...

10x=1.11111111...

10x-x=9x=1

x=1/9

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#18

(Original post by

Ok, I can see where you are coming from here. But I still think the infinities are the same because it has no end, I thought that was the whole idea of infinity? I'm also not trying to claim infinity as a number, just as a concept of unboundedness, so I didn't mean 0=1

I'm not very good at explaining things but I think this sum is similar to this question "show 1/9 =0.111111..."

Let x= 0.11111111...

10x=1.11111111...

10x-x=9x=1

x=1/9

**QuantumHatter**)Ok, I can see where you are coming from here. But I still think the infinities are the same because it has no end, I thought that was the whole idea of infinity? I'm also not trying to claim infinity as a number, just as a concept of unboundedness, so I didn't mean 0=1

I'm not very good at explaining things but I think this sum is similar to this question "show 1/9 =0.111111..."

Let x= 0.11111111...

10x=1.11111111...

10x-x=9x=1

x=1/9

And to the rational number thing.

It is not the same. Do you know why it is even 0.1111111...? Because of the decimal system. If you have another system, maybe dual system or any kind of, you won't have for 1/9 = 0.11111. It is because of the system you use.

And the main difference is, that you do not know I see, that @Noble. told already, that the main problem does not converge.

But your 0.1111... converges. Do you know what geometric series are?

If not check this out or google.

It just works because it converges, like I said. If it converges, then you can make equations because you have a result.

But "S1=1-1+1-1+... " does not converge. There is no result.

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#19

(Original post by

No, the inifinities are not the same, they can not be. As I showed you 1 message before. This is not the whole idea of infinity. You can not use infinity in normal mathematical equations. The axioms does not apply to infinity. It does not work.

And to the rational number thing.

It is not the same. Do you know why it is even 0.1111111...? Because of the decimal system. If you have another system, maybe dual system or any kind of, you won't have for 1/9 = 0.11111. It is because of the system you use.

And the main difference is, that you do not know I see, that @Noble. told already, that the main problem does not converge.

But your 0.1111... converges. Do you know what geometric series are?

If not check this out or google.

It just works because it converges, like I said. If it converges, then you can make equations because you have a result.

But "S1=1-1+1-1+... " does not converge. There is no result.

**sbj**)No, the inifinities are not the same, they can not be. As I showed you 1 message before. This is not the whole idea of infinity. You can not use infinity in normal mathematical equations. The axioms does not apply to infinity. It does not work.

And to the rational number thing.

It is not the same. Do you know why it is even 0.1111111...? Because of the decimal system. If you have another system, maybe dual system or any kind of, you won't have for 1/9 = 0.11111. It is because of the system you use.

And the main difference is, that you do not know I see, that @Noble. told already, that the main problem does not converge.

But your 0.1111... converges. Do you know what geometric series are?

If not check this out or google.

It just works because it converges, like I said. If it converges, then you can make equations because you have a result.

But "S1=1-1+1-1+... " does not converge. There is no result.

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#20

(Original post by

Ok, you've beaten me. I'm only an A level mathematician. http://math.arizona.edu/~cais/Papers/Expos/div.pdf This paper however continues the fight! What would you say to dispute the seemingly logical deductions here?

**QuantumHatter**)Ok, you've beaten me. I'm only an A level mathematician. http://math.arizona.edu/~cais/Papers/Expos/div.pdf This paper however continues the fight! What would you say to dispute the seemingly logical deductions here?

I don't have the time to go this through but in first case I'd say you can't do it because of rearrangement of the series. There he rearranges the sum.

But he proves there something.

So, I searched a bit and found the answer why it "works".

See here.

Although the full series may seem at first sight not to have any meaningful value,

**it can be manipulated**to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis, quantum field theory and string theory.
The series can be summed by zeta function regularization. When the real part of

*s*is greater than 1, the Riemann zeta function of*s*equals the sum . This sum diverges when the real part of*s*is less than or equal to 1, but when*s*= −1 then the analytic continuation of ζ(s) gives ζ(−1) as −1/12.And here you can understand why:

**Ramanujan summation**is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series,

**for which conventional summation is undefined.**

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