# Apparently 1+2+3+... = -1/12 (1 Viewer)

#### Sy123

##### This too shall pass
A recent Numberphile video had "proven" this result and during their proof they seem to run off the axiom that

$\bg_white S = 1 - 1 + 1- 1 + 1 - 1 + \dots = \frac{1}{2}$

And they said that this is the case since if we stop at an integer N it will be either 1 or 0 depending on the parity of N, so they said we just take the average of it....

Why is taking the average even legitimate?

#### seanieg89

##### Well-Known Member
A recent Numberphile video had "proven" this result and during their proof they seem to run off the axiom that

$\bg_white S = 1 - 1 + 1- 1 + 1 - 1 + \dots = \frac{1}{2}$

And they said that this is the case since if we stop at an integer N it will be either 1 or 0 depending on the parity of N, so they said we just take the average of it....

Why is taking the average even legitimate?
Firstly you should be careful with this video, it is pretty non-rigorous (as it has to be, being a pop maths channel). There is a lot being swept under the carpet with his manipulations.

Regarding your specific question: There are many notions of convergence other than the one you vaguely know in high school and rigorously pin down with epsilons and deltas in first year uni maths. For example, one such alternate notion is Cesaro summation. That alternating sum converges to 1/2 in the Cesaro sense, even if it is a divergent series in the "usual" sense.

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#### anomalousdecay

It depends really.

I'm going to say the below in an electronic perspective and it may have nothing to do with maths.

For example, quantum computers are based off qubits. Now, qubits are constantly on 1 and 0. So qubits are defined as being both at 1 and 0 at the same time. So this leads rise to how this is even possible. Now to prevent confusion in a situation where something can be both, the average is taken as this is only a definition, not something tangible.

Its like having the complex plane over the real plane. The complex plane can not be found in everyday life, unless it is necessary in a few strands of engineering and science. However, we use the real plane all the time. For example, we never pay $2i dollars for milk at the local shop. Instead we pay$2.

So its kinda like a definition rather than something you can get.

When programming softwares for such things, they will either spew out no answer, 1/2 or both 0 and 1 at the same time (or simultaneously). They will not spew out just a 0 or just a 1 as this leads to an indefinite argument. Logic gates would preferably spew out a 0 and 1 at the same time. This mixed signal would be averaged in the electronic circuit and you would get half of the full amount of a 1.

For example, let the signal for 1 be a voltage of 5V, and the 0 have a voltage of -5V.

In the end, the output signal would be 0V, which is thus the average of the two values.

However, this case only works for two values which are negative of each other (conjugates) or else the result would be a little different and would not exactly be the average (its some complex stuff to do with electronics).

I don't know whether what I just said above has anything to do with it mathematically, but I know that the above would be the case in electronics.

#### seanieg89

##### Well-Known Member
As for the sum of the natural numbers, that is an example of trying to extend the domain of convergence for a Dirichlet series. (This is similar to how we extend the Riemann zeta function to be defined on the whole complex plane excluding z=1.) The natural way to do this is by something called analytic continuation.

Some further reading if you are curious: http://terrytao.wordpress.com/2010/...tion-and-real-variable-analytic-continuation/. Might have to do a bit of wiki hyperlink chasing though.

Also check out the write-up by the guy in this video: http://www.nottingham.ac.uk/~ppzap4/response.html.

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#### RealiseNothing

##### what is that?It is Cowpea
For example, one such alternate notion is Cesaro summation. That alternating sum converges to 1/2 in the Cesaro sense, even if it is a divergent series in the "usual" sense.
ok b10

#### anomalousdecay

Is this concept similar to how you get infinite sums of things that tend to a definite number?

#### seanieg89

##### Well-Known Member
Is this concept similar to how you get infinite sums of things that tend to a definite number?
Not quite sure what you mean. It is an infinite series that "tends" to a definite value just like a geometric series with small ratio "tends" to a definite value. The former use of the word "tends" is just weaker.

#### anomalousdecay

Not quite sure what you mean. It is an infinite series that "tends" to a definite value just like a geometric series with small ratio "tends" to a definite value. The former use of the word "tends" is just weaker.
Yeah like one of these (I'm so bad at explaining maths using English, but can understand English using maths):

$\bg_white \sum \limits_{n=1}^{\infty} \left ( \frac{1}{2} \right ) ^{n} = 1$

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#### seanieg89

##### Well-Known Member
Yeah like one of these (I'm so bad at explaining maths using English, but can understand English using maths):

$\bg_white \sum \limits_{n=1}^{\infty} \left ( \frac{1}{2} \right ) ^{n} = 1$
Yeah, loosely similar. Convergence is an incredibly broad notion though.

#### anomalousdecay

Did you watch this one too Sy?

EDIT:

In the original video: Tony Padillia: "This might seem like mathematical hocus pocus, but we know its not because of Physics."

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#### anomalousdecay

Actually just watched that video now and its more like what I said before in my first post here.

#### asianese

##### Σ
Note that this was used in the context of string theory so a lot of mathematics is just taken as things appear to be : (Oh..it's -1/12 and if we assume the cessaro mean it works!! OMGZZ STRING FEORY!!)

In a mathematics sense, obviously it's not rigorous at all and is wrong.

#### seanieg89

##### Well-Known Member
Note that this was used in the context of string theory so a lot of mathematics is just taken as things appear to be : (Oh..it's -1/12 and if we assume the cessaro mean it works!! OMGZZ STRING FEORY!!)

In a mathematics sense, obviously it's not rigorous at all and is wrong.
That is pretty derogatory to physicists. This is just an attempt at relating an intricate topic to a non-specialist audience. (And a reasonably successful one judging by the number of non-maths friends I have heard talking about this video recently.)

There IS underlying rigour that cannot and should not be conveyed in a short youtube video. As for the bolded, "wrong" seems to imply that there is a "definitive" interpretation of that sum as a real number which is not equal to -1/12. I don't think there is any such interpretation.

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#### anomalousdecay

Note that this was used in the context of string theory so a lot of mathematics is just taken as things appear to be : (Oh..it's -1/12 and if we assume the cessaro mean it works!! OMGZZ STRING FEORY!!)

In a mathematics sense, obviously it's not rigorous at all and is wrong.
Yes I was thinking the same that it doesn't apply to maths completely, but rather to other stuff:

In the original video: Tony Padillia: "This might seem like mathematical hocus pocus, but we know its not because of Physics."
However, I didn't think of it in regards to string theory.

I thought of it in terms of digital format and it still somehow works.

#### Sy123

##### This too shall pass
So from what I understand its people who are fiddling with the definitions of summation and convergence in order to fit pre-assumed scientific theories for the purpose of saying "look our physics is mathematically valid!"?

#### anomalousdecay

So from what I understand its people who are fiddling with the definitions of summation and convergence in order to fit pre-assumed scientific theories for the purpose of saying "look our physics is mathematically valid!"?
I would say that for the purpose of digitalisation it works mathematically too as it is slightly different. So your statement won't apply to it.

However, for something like string theory, you are spot on the money with that definition there.

#### seanieg89

##### Well-Known Member
So from what I understand its people who are fiddling with the definitions of summation and convergence in order to fit pre-assumed scientific theories for the purpose of saying "look our physics is mathematically valid!"?
No, these definitions and concepts existed long before physicists found they modelled things that they observed well.

And I don't think that use of these concepts was artificially to gain "mathematical credibility". The maths just happens to be well suited to studying those concepts.

It just comes across a bit cheaper because the presentation was super colloquial so that some of the people who would normally stop watching after hearing the phrase "analytic continuation" would stay and watch the whole thing.