Infinity or -1/12?

David Berman and Marianne Freiberger




The maths


$S_1 = 1$

$S_2 = 1+2 = 3$

$S_3 = 1+2+3=6$





  \[ S_ n = 500,500, \]    

and for $n = 100,000$ you get

  \[ S_ n = 5,000,050,000. \]    

This is why mathematicians say that the sum



Srinivasa Ramanujan

Srinivasa Ramanujan






$S_1 = 1$












Extending the Euler zeta function










This is one way of making sense of Ramanujan’s mysterious expression.

The trick



is equal to $1/2.$



But this is just the original sum, implying

  \[ Z+Z=2Z = Z. \]    


In fact, you can derive all sorts of results messing around with infinite sums that diverge (see here). It's a trick!

The physics


Casimir effect

Illustration of the Casimir effect. Image: Emok.














About the authors


Marianne Freiberger is editor of Plus.


Dear David and Marianne,






Gerald Goldin
Rutgers University







Similar methods are used in the final proof.

There is no single unique "value" obtainable for a nonconvergent series through rearranging, regrouping, adding, etc. nonconvergent series. If you check out comments for the Numberphile video, you can find several different values produced for the same series, and although many of them are done incorrectly there are still several cases that correctly perform operations that would be allowed for convergent series. Once you have several different values all "equal" to the same thing, you can in principle derive all sorts of nonsense including 1=0.

You're not justified in shifting the terms over. You can't arbitrarily add 0-values to an infinite sum. And if you think about it conceptually you cannot move a line that extends to infinity, you would be stretching/compressing it.

Also to compare the functions S and S' vertically by adding 0 at the beginning of S', then you would need to add 0 at the end of S. This is obviously not possible.

Adding a zero doesn't change anything about a series. And he is justified in his calculations.

Thanks for your comments, I agree completely










Averaging the partial sums is the simplest case of what's known as Cesàro sums. Here's why that technique is attractive: If the original series converges to a particular value, so does its Cesàro sum; if the original series diverges, so does its Cesàro sum. For series that are neither convergent nor divergent, such as the Grandi series, sometimes the Cesàro sum produces a definite finite value. Some people (not I!) therefore think that the Cesàro sum can be used in place of the standard sum.



Thank for a great night read :)

Z = 1 - 1 + 1 - 1 + 1.....
Z' =0 +1 - 1 +1 - 1.....

Z+Z' => All but the 1st term cancel out, leaving just 1.
2Z = 1.
Z = 0.5

Or another way:


2Z = 2 - 2 + 2 - 2 + 2.....
So, 2Z is either 2 or 0; average 1.
2Z = 1.
Z = 0.5

1-1+1-1+1-1...+1-1+1-1+1... = 2Z.
It does not equal Z as the article states.



Can someone comment on this? I no longer believe anything!!!

Listen carefully in the explanation video. At one point, the man suggests the series s1 = 1 - 1 + 1 - 1 ... should be taken to be 1/2. The reasoning he gives for this is that "if we stop at an odd point the sum equals 1, if we stop at an even point then the sum equals 0". Both of those are true on their own, but hold on. If we "stop" at a point to observe a value, then that value must just based on that one part of the total sum. We can derive a value by averaging two repeating parts of our sum, but this means that the value is based on the parts, not the sum itself. the fact that the series 1 - 1 + 1 - 1 ... is cyclic (repeats the same pattern of numbers) allowed the presenter to derive a fact about all the parts at once, but this fact was not the sum of the series. The rest of their calculations are all derived from this one, and thus are all based upon facts about the parts that make up the sum, rather than what it equals, or "sums". This doesn't mean that the calculations are useless, though. These numbers represent facts about the sums that could prove useful.




OMG !! What Have YOU Done?????
Z+Z= the original sum i.e. Z
so you write 2Z=Z
so putting Z=0.5, we get 0.5=1
but why stop here?
so, Z=Z or Z=nZ

//OMG !! What Have YOU Done?????
Z+Z= the original sum i.e. Z
so you write 2Z=Z
so putting Z=0.5, we get 0.5=1
but why stop here?
so, Z=Z or Z=nZ
so whatever Z is, Z can be anything's nonsense. RIGHT??//

genius this is the exact point they also made :p really ROFL

This is the problem w infinite sums and calculus...people don't know all the rules, and make folly out of the work done by geniuses before them.

You cannot add infinites like numbers, because they're not numbers! We've found cheeky ways of representing infinity w finite numbers, but infinity , itself, is beyond what our finite minds can truly imagine.

Either way, when you Transpose the infinite concept into our finite world, finite operations work with the finite representation.

Assume infinite sum of numbers (1+2+3+4...) IS equal to -1/12 (we can assume this because we proved it is. Like we proved 2+2=4, or other convergent infinite sums). Let this equal Z
Using the new finite concept, Z+Z is -1/6, which is the same as 2Z..true!
Z+Z =/= Z+Z+Z...Z because you're mixing together the finite and infinite worlds.
2Z=/= nZ except when n=2. This is true using our finite point of view. Not with the infinite point of view. Remember, we only use Z to represent the finite value. If we don't take this value into consideration (we don't use proper infinity aka, we use a REALLY LARGE NUMBER), it breaks apart. Because we aren't using Z anymore.

Also, assuming S=1-1+1-1+1-1+... it's the same deal. Once you STOP, or if you don't use infinity properly, you will get a finite value, which is not infinite, and therefore not S. In the case of Z, if you stop early, you get a huge number, not -1/12. Same here, if you stop randomly, you get S=0 or 1, not 1/2. In infinite space, unfathomable to us, (finite things are unfathomable to us) it would converge (practically, since we're transposing it to our finite world, as with any convergent infinite sum).

Z+Z=Z. This works if only the first infinite series ends with a -1. But its infinite... doesn't end. so the argument is invalid

My apologies about this, my comment is not about the sums but about the casimir effect. You mentioned that classical physics says that there should be no net force between the plates. But my high-school science teacher told us that there is a gravitational force acting between any two masses.

There would be a gravitational force as well, but the unexpected result is an electromagnetic force. The plates are electrically neutral so you wouldn't expect any attraction, but the zero-point energy of the vacuum indeed draws them together/apart.

That's because the article is wrong, and little backdoors like this are tricking you into thinking the sum is infinity.

-1/12 is the correct answer.

I'm Einstein's ghost.

Thank you. One person who got it correct.


I think spreading ignorance will rather attract ignorant fan-boys and repel those with rational reasoning.

I disagree, because I've seen this video and got curious, I've found this article and now I understand.

I think they could have been a bit clearer that this is a sleight of hand though.

If Z(-1) = 1+2+3+4+5+....= -1/12. & Z(0) = 1/1^0 +1/2^0 +1/3^0 +1/4^0 +...
Z(0) = 1+1+1+1+1+...
[Z(-1) +Z(0)] = 2+3+4+5+... = 1+2+3+4+5+... -1= -1/12 -1 = -13/12
This implies Z(0) = -1
[Z(-1) - Z(0)] = 0 + (1+2+3+4+5...) = -1/12
This implies Z(0) = 0
Both these values for Z(0) of -1 & 0 are contradictory.
Adding [Z(-1) + Z(0)] + [Z(-1) - Z(0)] = -13/12 -1/2 = -14/12 = -7/6
This implies 2Z(-1) = -7/6 => Z(-1) = -7/12This contradicts Z(-1) = -1/12 which
was the original assumption.
Putting infinite divergent series equal to a finite number appears to lead to
results that are not sensible.

Consider these two divergent series.

Sp = +1 + 1 + 1 ....
Sn = -1 - 1 - 1 ...

If added together, all terms cancel and Sp + Sn = 0.

However, if one allows the illegitimate preliminary step of "shifting" either series left or right, one can arbitrarily produce a result of any integer value you want to get. The operation of shifting (or of defining a "series" that starts with a finite number of zeros) creates a finite series that becomes the result. The amount of shifting determines the number you get, because the rest of the infinite series will still cancel.

BTW, notice also that Sp + Sn = S1 from the video. So, if one rules out illegitimate cheating steps that artificially create a finite series (e.g. shifting or bracketing), the result is that Sp + Sn = S1 = 0.

Some of positive natural numbers is n(n+1)/2

So if we put n(n+1)/2 =-1/12 we get value of n less than 1.


Let's take our sum to infinity where the sum is supposed to be -1/12. Then let's add -1/12 to each side, and the sum is now zero. Am I missing something? Actually the whole edifice of this folly seems to be that these people treat infinity as if it were a number. Now if I can add 1/12 to infinity, it really wasn't infinity anyway. The series cannot converge unless there is a fudge factor, and that fudge factor is a game on paper, (and in some peoples' heads), that doesn't exist in nature (and logic). After all, nature doesn't work with infinitesimals: The shortest time is Planck time is 10**-43 seconds, and the shortest distance is 10**-35 metres. These are not because our tools are not good enough to measure them, it's because these are the smallest packets of time and distance that occur in nature. So the modern day versions of Zeno's paradoxes fail because they arbitrarily assign values to infinity and infinitesimals when required, and remove them when it doesn't suit the purpose. Infinity is still a theoretical, and non-real concept. If nature doesn't use it, then neither should we.

I must disagree with your assumption that we should not use anything not seen in nature in Mathematics. That may be a useful viewpoint in some sciences, but Mathematics often disconnects itself with reality only for the reality that belongs with it to be found later. In the 1500s, complex numbers were regarded as a ridiculous concept, yet now they are well accepted, their properties well documented, and natural cases have been found where they are used frequently. In electrical engineering, for example, heavy use is made of complex numbers. The concepts of infinitesimal and infinity are indeed quite necessary, as without it the idea of convergence and differentiation are less well-defined, perhaps impossible. So if your job, whether by its nature, its equipment, or otherwise relies on calculus, you do indeed need infinitesimals. Avogadro's number and the speed of light as values are solely decided by the measurements we use to find them, while π, e, γ, ρ, and many other Mathematical constant's don't change, regardless of measurement.

Your understanding of Planck units is flawed. Planck unit are merely a system of natural units in which G, c, and h-bar have a value of 1. Nothing in physics prevents values smaller than Planck units. For instance, calculate the Planck energy unit. It is huge compared to the energy of the battery in my cell phone, or the energy of a photon of visible light.

Extreme care is needed when dealing with infinite sums. From what I can tell, it seems that confusion arises when people assume they are allowed to manipulate infinite sums in the same way as they are used to doing with the finite sums they are accustomed to dealing with. Assuming infinite sums behave anything like finite sums will lead to problems. Reordering a non-convergent infinite sum will often produce different answers - in many cases you can get any result you desire, but it's clearly incorrect. Proceeding term-by-term to calculate the partial sums. What does it even mean to sum an infinite number of terms when the sum is non-convergent? It's unsafe to assume that what works for finite sums must apply to infinite sums as well.

Regarding -1 + -1 ... (Grandi's Series), although it doesn't have a limit, it is Cesaro summable and as a cesaro sum of 1/2. This can be easily proved in lots of ways. This is what Numberphile were talking about no doubt.

I was pretty blow away at that stage of the "proof." Let's just say the infinite series 1-1+1-1+1-1+1-1... is solved by averaging 0 and 1? what? There may be an explanation but that is not it.

They said that for the sake of time. If you re-watch the video you'll see a link to a previous video dealing with that series in greater detail.

This website hammers down on the first sum Numberphile uses, 1-1+1-1+...
This is called Grandi's series, look it up anywhere and you will find the answer IS 1/2, because of simple algebra.
People are always like "You can't use algebra on infinities", I have to respectfully disagree. In order to figure out infinities, you treat them like variables. The only difference between infinity and x is infinity can't equal something, x can. So when dealing with sums, we can treat infinity like a variable and figure out the sum from there. This is pretty much what Numberphile did the entire video excluding Grandi's series, which can be proven multiple ways elsewhere on the internet.

In the sum, substitute 1+1 for 2, 1+1+1 for 3, etc to get 1+1+1..., now do it with the even numbers, then the odd, then the primes, all result in 1+1+1 forever. Do they all equal -1/12?

What about following sum?

-infinity.........+-3 + -2 + -1 + 0 + 1 + 2 + 3 .........+infinity

I guess answer to this is -1/12 :)

1+2+3+4+5+....+infinity = infinity

You can only play around by switching positions in a series if you consider both ends because its infinite series..

Another you-tuber mathologer proved this in his video but I somehow still don't get it. Here's the link:

So correct me if I'm wrong.
Can't we write Z = 1-1+1-1+1 as this?>>>>> (1+1+1+1+1+.....) - (1+1+1+1+1+1+....) = Z
All I did was pair up the positive 1's and the negative 1's and added them together.
If you assume that both sets of sums add up to infinity, their result would come out to be zero.
But, you can also look at what the sum of 1+1+1+1.... was proved to be. Which is -1/2. So in this sense,
(-1/2) - (-1/2) = (-1/2) + (1/2) = 0. So Didn't I just prove that the sum 1-1+1-1+1.... = 0?
If not, where is my mistake?

I think, rearranging (1-1+1-1+1-1...) as (1+1+1+1...) - (1+1+1+1...) is not valid because of the "shifting" of values, and decomposing of one infinity into two. If you insist on "proving" equality to 0 - there is an easier way: just use parenthesis like this: (1-1)+(1-1)+(1-1)... = 0 + 0 + 0 ... which is "clearly" zero. Right? Not really.

This has been bugging me all day, and the best "intuitive" explanation may be based in physics: If you draw (1-1+1-1+1-1...) on a graph assuming it's some physical value over time - it's easy to see that 1/2 is the center of oscillation. So, even though the graph never converges to 1/2 - in the infinity it may as well converge. Given the example with a light bulb.. which is ON or OFF, in the infinity, the bulb would be neither ON or OFF - it would be half-bright. If you start sequence with -1, you get an oscillating line around -1/2. So, that makes sense too. If you start doing tricks like (1-1)+(1-1)+(1-1)... = 0 + 0 + 0 ... -- it's easy to see that the trick here is selectively collapsing time intervals to 0, which doesn't make sense in the physical sense.

I started today thinking that (1-1+1-1+1-1...) = 1/2 was a fallacy, but now I think it's actually true, and it starts to make sense. However, I'm still to make the leap to understanding how (1+2+3+4+5+...) = -1/12 can be a useful fact, even if it's mathematically correct.

consider the geometric series, with |x|<1:
as x goes to -1 from right, S goes to 1/2.

the answer in numberphile is not wrong. you are only considering it in terms of finite maths axioms. when we analyze it in terms of an infinite series we do indeed get -1/12. this makes perfect sense and can be graphed. no magic need be used.

Just got home from work to have my 18 yr old son tell me about the numberphile video proof. We watched the video and I told him immediately that the false step was stating that 1-1+1-1... = 1/2. That series sum fluctuates between 0 and 1 and can never equal 1/2. The average is 1/2. Thanks for posting this article because he was falling for the magician's trick.

There is proof that 1-1+1-1+1-1....= 1/2. A simple google search will show you that this is Grandi's series and it is accepted to be 1/2. Proof is everywhere to be found. It's not a "magician's trick".