The Sum of All the Natural Numbers

[AlfieD]

Hi lovely people,

I recently came across a video http://www.youtube.com/watch?v=w-I6XTVZXww that said if you add all of the natural numbers from 1 to infinity, the answer is... What do you think it is? Infinity or something like that?

They said it was -1/12. I watched the proof but I don't understand the logic behind it because if you add positive numbers together, how can you get a negative? And if you're adding whole numbers together, how can you get a fraction (not like 5=5/1, you know what I mean)?

I heard them say that it's essential to String Theory and all 26 dimensions coming out. I assume they meant bosonic string theory? Because that's the only subset of string theory I know that has 26 dimensions, I'm pretty sure the original has 10.

I'm hoping that someone can shed some light on this so called 'astounding result'.

Thanks in advance,
AlfieD

 

[mfb]

This is not the sum of all natural numbers. He uses formulas in a region where they do not apply.

As a comparison: "all odd numbers between 2 and 8 are prime" is true (as 3, 5 and 7 are prime) and easy to prove, but that does not mean 1 or 9 would be prime as well because they are not in the region where the statement is usable.

 

[AlfieD]

This is not the sum of all natural numbers.

In what way exactly? It says that it's the sum of all the positive whole numbers from 1 to infinity.

 

[1MileCrash]

This is not the sum of all natural numbers. He uses formulas in a region where they do not apply.

As a comparison: "all odd numbers between 2 and 8 are prime" is true (as 3, 5 and 7 are prime) and easy to prove, but that does not mean 1 or 9 would be prime as well because they are not in the region where the statement is usable.

Did you watch the video? Grimes proved it without involving any type of formula, he just did algebra with other series.

 

[mfb]

@AlfieD: Yeah sure, but it is wrong. Entertaining, but wrong.

With incorrect manipulations of infinite sums, you can prove anything.As an example, consider the sum 1/2 - 1/3 + 1/4 - 1/5 + 1/6 - 1/7 +-...
Clearly the sum is positive everywhere and increases every two steps, so it is larger than zero.
1/2 - 1/3 + 1/4 - 1/5 + 1/6 - 1/7 +- ... > 0
Let's rearrange it a bit:
- 1/3 - 1/5 + 1/2 - 1/7 - 1/9 + 1/4 +- ...
Now -1/3 - 1/5 = -8/15 and 8/15>8/16=1/2, and in the same way -1/7 - 1/9 = 16/63 > 16/64 = 1/4 and so on.
Therefore, this sum is clearly negative.
- 1/3 - 1/5 + 1/2 - 1/7 - 1/9 + 1/4 +- ... < 0

How can the same sum be larger and smaller than zero at the same time?

Well, the answer is bad mathematics - this rearrangement is not valid, it changes the value of the sum. The same is true for the steps made in the video - they are just not valid mathematics.

@1MileCrash: Yes I watched it. Maybe "formulas" is not the best word, let's say "calculations".

 

[1MileCrash]

@1MileCrash: Yes I watched it. Maybe "formulas" is not the best word, let's say "calculations".

OK, you said "in regions where they do not apply" so I thought it sounded like you were referring to Riemann-Zeta regularization, which was not involved.

I'm not very convinced there there was anything wrong with the work he did, but I will look at what you said more closely.

 

[AlfieD]

@AlfieD: Yeah sure, but it is wrong. Entertaining, but wrong. [text...] - this rearrangement is not valid, it changes the value of the sum. The same is true for the steps made in the video - they are just not valid mathematics.

This is essential to bosonic string theory in allowing the 26 dimensions to exist, so are you saying that bosonic string theory can't be correct?

 

[AlfieD]

OK, you said "in regions where they do not apply" so I thought it sounded like you were referring to Riemann-Zeta regularization, which was not involved.

Yeah, that Riemann-Zeta thingy was in a different proof, but as you said, it wasn't involved in the linked video proof.

 

[Student100]

http://www.youtube.com/watch?v=E-d9mgo8FGk&feature=youtu.be

Is their other 'proof' video, but it's still incorrect.

The first video is just mathematical hand waving.

They're misusing the mathematics.

 

[DrewD]

OK, you said "in regions where they do not apply" so I thought it sounded like you were referring to Riemann-Zeta regularization, which was not involved.

I'm not very convinced there there was anything wrong with the work he did, but I will look at what you said more closely.

I don't know if "wrong" would be the exact term that I would use, but it is definitely misleading. They are using and rearranging divergent sums to get a result. There is a sense in which it is true (from what I have read, but I am not an expert in this area), but what he does is not valid using standard summation. Even from the beginning they are breaking "rules". For example $\Sigma^{\infty}_n(-1)^n$ does not converge. Sure, it can be useful to redefine a sum as the average of the partial sums (which I am pretty sure is what he is doing), but it is misleading to say that this holds true when we are considering a standard summation.

He also rearranges the sums for his second sum (I think-I have watched the video, but not today). Since an infinite sum is usually defined as the limit of the sequence of partial sums, this is not acceptable.

This is not something that I know much about, but I have been told that the truth of this statement is important to some conclusions in math and physics, but his "proof" is extremely flawed.

 

[1MileCrash]

I don't know if "wrong" would be the exact term that I would use, but it is definitely misleading. They are using and rearranging divergent sums to get a result. There is a sense in which it is true (from what I have read, but I am not an expert in this area), but what he does is not valid using standard summation. Even from the beginning they are breaking "rules". For example $\Sigma^{\infty}_n(-1)^n$ does not converge. Sure, it can be useful to redefine a sum as the average of the partial sums (which I am pretty sure is what he is doing), but it is misleading to say that this holds true when we are considering a standard summation.

He also rearranges the sums for his second sum (I think-I have watched the video, but not today). Since an infinite sum is usually defined as the limit of the sequence of partial sums, this is not acceptable.

This is not something that I know much about, but I have been told that the truth of this statement is important to some conclusions in math and physics, but his "proof" is extremely flawed.

1 - 1 + 1 - 1 +... does not converge by our definitions, but the argument that it is 1/2 is convincing to me. I'm not so sure that our standard notion of convergence is the only way to associate a series with a value.

 

[AlfieD]

Can I just ask whether the quarrel anyone has is with this particular proof and not the actual answer itself? So, are you fine with it being -1/12, but you just think that the proof is highly floored. If so, do you know of any better proofs?

 

[Student100]

It's only -1/12 when used in the context of ζ(-1). Suggesting that the infinite sum 1+2+3+4+5... is -1/12 is just wrong, it's an abuse of notation.

Someone correct me if I'm wrong, I'm working off a limited knowledge base here.

 

[mfb]

I'm not very convinced there there was anything wrong with the work he did, but I will look at what you said more closely.

None of the sums he uses has a proper value.

As another example:

Assume 1+2+3+... = -1/12.
Then clearly
0+1+2+3+... = -1/12.
Taking the difference:
1+1+1+... = 0.
In the same way,
0+1+1+1+... = 0.
Taking the difference again,
1+0+0+..=0
1=0

Wait... no.

This is essential to bosonic string theory in allowing the 26 dimensions to exist, so are you saying that bosonic string theory can't be correct?

Please give a reference that the actual sum of natural numbers (and not the value of the Riemann Zeta function) is used there.

 

[AlfieD]

Please give a reference that the actual sum of natural numbers (and not the value of the Riemann Zeta function) is used there.

Ummm, if you watch the video the guy shows you the sum inside a book entitled String Theory. Listen closely to what he says as well. I'm pretty sure it's near the start of the video.

 

[Mandelbroth]

Hi lovely people,

I recently came across a video http://www.youtube.com/watch?v=w-I6XTVZXww that said if you add all of the natural numbers from 1 to infinity, the answer is... What do you think it is? Infinity or something like that?

They said it was -1/12. I watched the proof but I don't understand the logic behind it because if you add positive numbers together, how can you get a negative? And if you're adding whole numbers together, how can you get a fraction (not like 5=5/1, you know what I mean)?

I heard them say that it's essential to String Theory and all 26 dimensions coming out. I assume they meant bosonic string theory? Because that's the only subset of string theory I know that has 26 dimensions, I'm pretty sure the original has 10.

I'm hoping that someone can shed some light on this so called 'astounding result'.

Thanks in advance,
AlfieD

I haven't watched the video yet, but I'm pretty sure that's using something called a Ramanujan summation. The idea was created by Ramanujan, a famous Indian mathematician. It's not an "actual" summation, but it can apparently be helpful sometimes in number theory.

Edit: Perhaps that's what the idea is, but in the video all I see is mathematical crackpottery. It hurts my eyes.

 

[DrewD]

Ummm, if you watch the video the guy shows you the sum inside a book entitled String Theory. Listen closely to what he says as well. I'm pretty sure it's near the start of the video.

Be careful. Have you read the book? I am looking for this part of the book, but I bet mfb is right: the book is probably using a value of the analytic extension of the zeta function. This is different (in ways the mfb is probably more comfortable with than I am) than just saying that adding all of the natural numbers together gives this value. It is extremely simple to show that this series, using the normal definitions of summation, is divergent.

The formula is not wrong when it is viewed properly, but the video is ambiguous and uses false mathematics for a "proof" that is meaningless.

 

[atyy]

Although the proof is wrong, some mathematicians have savoured it. In particular, Atiyah writes that the erroneous summation was Euler's, and that it was only in relatively recent times that we understand what Euler's intuition was pointing towards.

Another good fun source that discusses the history of this equation is John Baez's
My Favorite Numbers: 24
http://math.ucr.edu/home/baez/numbers/24.pdf


For Atiyah's comment see his article "How research is carried out" in http://books.google.com/books?id=YJ0cZwxLECAC&source=gbs_navlinks_s (p213).

 

[WannabeNewton]

I recently came across a video http://www.youtube.com/watch?v=w-I6XTVZXww that said if you add all of the natural numbers from 1 to infinity, the answer is... What do you think it is? Infinity or something like that?

One of my friends posted the exact same video on his facebook wall the same day you made this thread...O.O

 

[AlfieD]

One of my friends posted the exact same video on his facebook wall the same day you made this thread...O.O

Hmmm, that's strange. Haha.

 

[AlfieD]

Be careful. Have you read the book? I am looking for this part of the book, but I bet mfb is right: the book is probably using a value of the analytic extension of the zeta function. This is different (in ways the mfb is probably more comfortable with than I am) than just saying that adding all of the natural numbers together gives this value. It is extremely simple to show that this series, using the normal definitions of summation, is divergent.

The formula is not wrong when it is viewed properly, but the video is ambiguous and uses false mathematics for a "proof" that is meaningless.

In the book it looks like this: $\Sigma$^∞_n=-1/12

Although I can't get it to look right using the formatting available to me. The ∞ should be sitting on top of the Sigma, and the n should be adjacent to it. Then there's also an n=1 that is just beneath the Sigma. Obviously this is all followed by an equals sign and the -1/12.

 

[mfb]

Ummm, if you watch the video the guy shows you the sum inside a book entitled String Theory. Listen closely to what he says as well. I'm pretty sure it's near the start of the video.

See the following equations. They are not actually calculating the sum, they modify it ("insert a smooth cutoff factor", visible at 0:50) to something different.

You cannot actually sum the natural numbers, see my previous post for a proof that those calculations do not work (they lead to 1=0 which is certainly wrong).

 

[LCKurtz]

I haven't watched that video, but I will throw in my 2 cents worth anyway. There are ways to generalize the notion of convergence of a sequence so that ordinarily divergent sequences can be considered to be convergent. The divergent sequence $\{a_n\} = 1,0,1,0,1,0,...$ (pardon the abuse of notation) can be thought of converging "on the average" to $1/2$. This can be formalized by letting$$
s_n =\frac{a_1+a_2+...+a_n}{n}$$It is a common exercise in analysis books to show that if $a_n\to L$ then $s_n\to L$. In the above example, $s_n\to \frac 1 2$ so the sequence {$a_n$} converges "on the average" to $\frac 1 2$. This transformation can be viewed as multiplying the $\{a_n\}$ sequence by the infinite matrix$$
\left(\begin{array}{cccc}
1 & 0 & 0 &...\\
\frac 1 2 & \frac 1 2 & 0 &...\\
\frac 1 3 & \frac 1 3 & \frac 1 3 & ...\\
... & ... & ... & ...
\end{array}\right )$$to get the sequence$\{s_n\}$. Matrices like these give "summability methods" and they generalize the notion of convergent sequences to larger classes. If they preserve the notion of convergent sequences, they are called "convergence preserving summability methods". Google that for more information if you like. If they in addition preserve the limits of convergent sequences, as in the above example, they are called "regular" summability methods. You might also Google "toeplitz theorems" if you are interested in what characterizes regular methods.

My point is divergent sequences can be made convergent in a more general sense giving limits that might seem not to make sense. How appropos this is to the subject in the OP I don't know, but I wouldn't dismiss it out of hand.

 

[bahamagreen]

I'm not a mathematician...

If I didn't make a typing error, you can see that A through G look like they might be all representations of "the same thing", yet depending on how you group the numbers, it can look like the sum might be 0 or 1, or that a positive integer multiple of C or E might still be 0, but applied to B or F might be that positive integer... assuming it is proper to even think this way.

A= 1 -1 + 1 - 1 +... = ?
B= 1 + (-1 + 1) + (-1 + 1) +... = 1 + 0 + 0 + 0 +... = 1?
C= (1-1) + (1-1) +... = 0 + 0 + 0 +... = 0?
D= 1 + (-1 + 1 -1) + (1 -1 +1) +... = 1 - 1 + 1 -1 +... = back to A?
E= (1 -1 +1) + (-1 +1 -1) +... = 2 + (-2) +... = back to A times 2?
F = 1 + (-1 +1 -1 +1) + (-1 +1 -1 +1) +... = 1 + 0 + 0 + 0 +... = B? =C +1? = 1?
and
G = 1 + (-1 +1) + (-1 +1 -1) + (1 -1 +1 -1) + (1 -1 +1 -1 +1) +... = 1 +0 + (-1) + 0 + 1... = A with extra 0s, so back to "zero fattened" A?

These don't seem well behaved... Are there special rules for grouping the elements of an infinite series, or are these things not allowed to be summed by various grouping algorithms?

Is there a "proper form" for these things or any allowed/disallowed conventions for grouping, intending to infer or evaluate the sum? Or are things like this considered indeterminate?

LCKurtz, I'm unable to follow your demonstration (my ignorance), but if a method works for one representation of a thing, but another representation of the same thing is indeterminate, are they representing the same thing? Or is the indeterminate representation simply incomplete or flawed in principle?

 

[mfb]

My point is divergent sequences can be made convergent in a more general sense giving limits that might seem not to make sense. How appropos this is to the subject in the OP I don't know, but I wouldn't dismiss it out of hand.

It is possible, but then you have to introduce (and define) this method - something that is not done in the video.@bahamagreen: Check your sequence E, 1-1+1=1 not 2.

 

[lendav_rott]

The sum of all natural numbers on to infinity? Well, that's strike 1.
Substituting S to contain the sum of all natural numbers INCLUDING infinity, implying it is a natural number, and calling it a real figure doing real figure maths with it. Strike 2.
Somehow managing to subtract infinity from infinity - Strike 3 - you're out.

 

[PeroK]

It would be interesting to know what these things are, given that the string-theory physicists seem to think they have something more than a dodgy paradox here.

At the root of the issue, as ever, is the assumption that if you say something is a number it must be a number. If:

$\sum_{n=1}^{∞}(-1)^{n+1} = 1/2$

$\sum_{n=1}^{∞}(-1)^{n+1}n = 1/4$

$\sum_{n=1}^{∞}n = -1/12$

Then, these symbols no longer indicate the regular real numbers, the normal process of addition and the regular meaning of $\sum_{n=1}^{∞}$

Note that the second formula does not meet the naive criterion presented in the video for the sum of an infinite series being the average of its partial sums. So, if the justification for the first sum is applied to the second, the thing already breaks down.

If the string theorists are not joking, then there must be something in string theory that sort of looks like a number; something that sort of looks like addition and some sort of concept of an infinite sum that leads to these results.

It would be interesting to know for what objects these answers make sense.

A very simplistic analogy of what (I suspect) they are doing is modulo arithemtic presented without the modulo.

1 + 1 = 0 (mod2)

But that doesn't mean that 1 + 1 really does = 0.

 

[lendav_rott]

After watching that video over and over and thinking about, there is just no explanation. At best what it proves is infinities are infinite and UNdefined. The "all natural numbers" is an arithmetical sequence. The sequence only has a sum if its elements and the amount of elements are finite. Suppose you go from 1,2,3,4... to 1 000 000 - then you can add them together and say it's a sum. But as n approaches infinity, it no longer is finite, therefore it does not have a sum to begin with.

 

[DrewD]

I wouldn't be so quick to question string theorists (about this). I think they are trying to "prove" this this way because you don't need a math degree to follow what they are doing. I think they do us all a disservice by presenting it this way, but I am sure that the use in string theory is justified. Look up the formula and string theory on wiki and it gives more depth. I don't know enough about string theory to know why it comes up.

 

[Student100]

This is also used in the Casimir effect from QFT, not just string nonsense. Its called like it is in all the descriptions I've read which is that they introduce a regulator into the summations to make the sums finite and then take the limit to remove the regulator. It's all a form of analytic continuation.

These guys are just trying to obscure all of this and make it seem legit to their viewers on youtube without telling the whole story.

 

[1MileCrash]

These guys are just trying to obscure all of this and make it seem legit to their viewers on youtube without telling the whole story.

That's kind of the impression I got too. They are doing a lot of viewer baiting.

What is really telling is that, even though I think the -1/12 result is indicative of something and that it shouldn't be dismissed, at the beginning of the video, the mathematician asked the camera man what he thought the sum was equal to. The camera man said "I would say it tends to infinity" and the mathematician said "no."

It's one of those "woah, the obvious answer is actually wrong!" hooks, except the obvious answer is not wrong regardless of your opinion of the -1/12 result.

 

[FlexGunship]

The fact these these techniques for "taming" a divergent series show up in physical (measurable) reality is convincing enough for me. Perhaps it's the type of thing you avoid in Algebra 1, but there's SOMETHING here. SOMETHING about this sum likes to be $-\frac{1}{12}$ in mathematics and in reality.

It seems (after further reading) that every method we have of calculating series summation (zeta, Ramanujan, algebraic) all lead to this result.

Dismissing it as "mathematical hand-waving" doesn't do it justice.

 

[Office_Shredder]

Flex, consider the following. In the second video you posted in General Discussion they showed
$$S_1 = 1 - 1 + 1 - 1 + 1 -1... = 1/2$$
$$S_2 = 1-2+3-4+5-6... = 1/4$$From here, consider
$$S_2 + S_2$$
in the following way: Take the second S₂ and bump it over by three spaces when lining it up under the first, so you get after canceling vertically
1 -2 + 3 - 4 + 5 -6...
_________1 - 2 + 3...
$$1 -2 +3 -3 -3 -3 -3... = 1/2$$
$$2 -3(1+1+1+1+...) = 1/2$$
solving gives
$$1+1+1+1+... = 1/2$$
Now consider
1+2+3+4+... - (1+1+1+1+...) by canceling the first, second, third etc. terms to get
0+1+2+3+4+...
From which we have proven that
S - 1/2 = S
So -1/2 = 0. Doing only the same thing they were doing in the video, only very slightly differently (aligning the $S_2+S_2$ step two steps further to the right).

The fact that there exists a set of invalid steps that appears to give the correct answer doesn't mean that those steps themselves are profound. The physics statement is "there is a continuation of the zeta function to s=-1 satisfying certain properties that I want for physics reasons, and the value at s=-1 is -1/12" which is cool, but when you try to reword it as "the sum of all natural numbers is -1/12, and I can prove it with algebra" you go from doing science and mathematics to proving that 1=0.

 

[PeroK]

The fact that there exists a set of invalid steps that appears to give the correct answer doesn't mean that those steps themselves are profound.

That about sums it up!

 

[MathJakob]

It just seems silly to me to say "We don't know what the last digit of 1-1+1-1+1... is so we'll just go in the middle and say the sum is a half..." Surely that sum should be undefined? If we don't know what the last digit is then we can't just say "oh what the hell we'll take the average and call that the answer..."

I don't know how often this sort of stuff is accepted in math but it just seems like extremely bad mathematics. It might work in equations ect but that doesn't mean to say that it actually does equal a half. Just like 0.999 recurring doesn't actually equal 1 because somewhere from thin air you're getting a extra 1...

If 0.999 recurring equals 1 then 0.34999 reucurring equals 0.35 I think it's just for sake of keeping things neat? I don't know I'm not nearly experienced enough to comment lol but I just thought I'd share why I think it's incorrect to say such things.