# The Sum of All the Natural Numbers

1. Jan 10, 2014

### 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'.

AlfieD

2. Jan 10, 2014

### Staff: Mentor

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.

3. Jan 10, 2014

### AlfieD

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

4. Jan 10, 2014

### 1MileCrash

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

5. Jan 10, 2014

### Staff: Mentor

@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".

6. Jan 10, 2014

### 1MileCrash

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.

Last edited: Jan 10, 2014
7. Jan 10, 2014

### AlfieD

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?

8. Jan 10, 2014

### AlfieD

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

9. Jan 10, 2014

### Student100

10. Jan 10, 2014

### DrewD

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.

11. Jan 10, 2014

### 1MileCrash

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.

12. Jan 10, 2014

### 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?

13. Jan 10, 2014

### 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.

14. Jan 10, 2014

### Staff: Mentor

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.

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

15. Jan 10, 2014

### AlfieD

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.

16. Jan 10, 2014

### Mandelbroth

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.

Last edited: Jan 10, 2014
17. Jan 10, 2014

### DrewD

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.

18. Jan 10, 2014

### 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).

Last edited by a moderator: Sep 25, 2014
19. Jan 10, 2014

### WannabeNewton

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

20. Jan 11, 2014

### AlfieD

Hmmm, that's strange. Haha.

21. Jan 11, 2014

### AlfieD

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.

22. Jan 11, 2014

### Staff: Mentor

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).

23. Jan 11, 2014

### 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.

Last edited: Jan 11, 2014
24. Jan 12, 2014

### 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?

Last edited: Jan 12, 2014
25. Jan 12, 2014

### Staff: Mentor

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.