# 1+2+3+4+...=-1/12 weirdness

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1. Jun 5, 2015

### zachx

I've recently accepted (reluctantly) that 1+2+3+4+...=-1/12, but wouldn't that mean the limit as x→∞ of f(x)=x not be ∞ but some maybe negative number because if it were ∞ then 1+2+3+4+... should also be ∞. Also, do numbers get so positive they become negative lol.

2. Jun 5, 2015

Staff Emeritus
Well you shouldn't. That series is divergent.

3. Jun 5, 2015

### certainly

It's actually quite useful in the study of divergent series and is called Ramanujan Summation.

4. Jun 5, 2015

### zachx

I thought that too, but this guy pulled out some string theory textbook.

5. Jun 5, 2015

### zachx

Oh, never mind thanks for clearing it up for me, but I still don't quiet understand why it's negative n

6. Jun 5, 2015

### certainly

Is that why you've been calling yourself a cat :-)
In regards to your question, it's a different type of summation, so I don't know how to make sense of it. (i.e. in the traditional sense the sum is still divergent).
I found an old thread that has a post by yasiru89 with a more formal derivation of it, may help.

7. Jun 5, 2015

### zachx

Moo.

Alright, well 1 last question: so when you multiply, you're adding a bunch of times and when you integrate you're finding the area, so conceptually what are you doing with the Ramanujan Summation?

8. Jun 5, 2015

### certainly

I don't think a geometrical interpretation exists. (if anyone knows of any, kindly point them out). However several analytical proofs exist.
[For instance you could use the relation between the zeta function and the Dirichlet eta function $(1-2^{1-s})\zeta (s)=\eta (s)$ and then Abel sum $\eta (-1)$.]

Last edited: Jun 5, 2015
9. Jun 5, 2015

### muzialis

I think the "geometrical interpretation" is the fact that the Ramahujan summation gives the first term in an asymptotic series related to the divergent sum. The matter is explained best here, https://terrytao.wordpress.com/2010...tion-and-real-variable-analytic-continuation/.
Once this is understood, even the physical application of the divergent sum regularisation (for example, in estimating the Casimir effect) make some sense.
There are other cases where divergent sums are useful in practice: for example Stirling's approximation of the factorial. It is given by a divergent series, but, fixing the number of terms in the series, the approximation for very large argument is very good (in a sense made precise by the definition of asymptotic series).

10. Jun 5, 2015

### micromass

Staff Emeritus
God, I hate that video. The video is very misleading. I hoped they would be somewhat clear in it.

First of all, the series $1+2+3+4+...$ diverges. You will find no mathematician that disagrees with this. The most natural sum is $1+2+3+4+... = +\infty$.
Now, what is the $-1/12$ thing all about? Well, some mathematicians have found a way to associate a number to divergent series. I would not call that number the "sum" of the series, it is just a number associated to it. In this case, the number associated to $1+2+3+4+...$ is $-1/12$. Now, we often write $1+2+3+4+5+... = -1/12$, but that's where you should be careful, since that $=$ sign does not mean the classical one, in fact it means that we evaluate the series in a nonstandard way (like Ramanujan summation). Now in many circumstances, replacing $1+2+3+4+...$ with $-1/12$ is wrong and a very bad idea, but in some it might work out. It should then be shown why exactly we can replace the sum by $-1/12$.

11. Jun 5, 2015

### Staff: Mentor

Oh thank god. That video was driving me crazy. I was telling myself, "There's no way that's right!".

12. Jun 5, 2015

### micromass

Staff Emeritus
That video is the equivalent of a teacher telling a high school class that there is a number whose square is $-1$, without saying that it is not a real number and without saying that we are essentially inventing a new number system. Sure, it will get you likes because it is mindblowing, but it is pedagogically awful.

13. Jun 5, 2015

Staff Emeritus
I agree with micromass, the problem is the equals sign, because it doesn't mean, well, "equals". Maybe a different symbol, like a double-ended arrow, would be less confusing. Or a functional that took a series as input.

14. Jun 6, 2015

### phion

Made me grin like an idiot.

15. Jun 7, 2015

### wabbit

16. Jun 21, 2015

### hunt_mat

It's to do with the analytic continuation of the Riemann zeta function. The things they do in that video are slight of hand (typical physicists) and so you shouldn't take their derivation seriously at all.

I was taught by Ed Copeland at university.

17. Jun 24, 2015

### davidmoore63@y

I've always thought the easier example is
X= 1+2+4+8+16+32+..
2x= 2+4+8+16+32+...
Subtract
-x= 1
X=-1
Nice

18. Jun 24, 2015

### Staff: Mentor

The main problem here is that you are doing arithmetic with two series that obviously diverge. In effect the arithmetic is leading one to believe that the indeterminate form [∞ - ∞] gives a meaningful answer.

19. Jun 24, 2015

### micromass

Staff Emeritus
But in fact, if the series converge, then the answer is correct. That is a big if of course, since in the usual real numbers, it doesn't converge. But in the 2-adic numbers, it does! And there we indeed have that $-1 = 1 +2 + 4 + 8 + ...$. But this is a completely different thing that the standard reals!

20. Jun 28, 2015

### thelema418

Is there a better video available on that topic? I am curious because high school teachers have told me that they use that video in their classroom.