(If you'd rather just know what 1+2+3+4+ . . .actually is equal to, check out our next post in this series)Brief Summary: 1+2+3+4+ . . . is not equal to -1/12, but both the infinite series and the negative number are associated with each other in a way that can be seen in this graph
The area of the little region below the horizontal axis equals -1/12, and the infinite area under the curve on the right gives you 1+2+3+4+. . . , which goes to infinity as you add terms, not to -1/12. (Update 2-6-14: of course, this graph shows what it looks like if you can only see a finite region of the graph. So 1+2+3+4+...+m is going to infinity, provided m is some finite number. This doesn't say anything about what happens if you could see all the way to infinity - I'd need a much bigger screen for that. For this reason, I am crossing out all the things that I am not so sure about in this post.)
That's a relief - the world makes sense again.
For a longer explanation, read on . . .
(First of all, I want to say I'm sorry to all the mathematicians whose heads exploded when I used non-mathematical reasons to justify why I believed 1+2+3+4+ . . . =-1/12. I was wrong, and after doing some actual math and pondering pretty pictures, I see why.
Thanks in large part to the patience and persistence of people like Bernd Jantzen, who commented extensively on my previous post
on this subject, I have found a way to simply, and graphically explain how the series 1+2+3+4+ . . . is associated with (and not equal to) the number -1/12, (update 2-6-14: provided you are actually taking the limit as you add terms and not looking at all the infinite terms at the same time as the Numberphiles did).
I usually try to avoid writing equations on this blog, so you're going to have to bear with me. But at least there will be pictures, and those are the most important parts.
Here we go . . .
I find it easier to understand things visually, so it occurred to me to plot out the series 1+2+3+4+ . . . at various points as you add the numbers. These points are called partial sums. This is what you get if you stop the sum after each step
=1 the partial sum is 1
=2 the partial sum is 1+2 = 3
=3 the partial sum is 1+2+3=6
=4 the partial sum is 1+2+3+4=10
=5 the partial sum is 1+2+3+4+5=15
If you draw the first five terms on a piece of paper, it looks like this
But you don't have to add all those numbers to calculate value at each point. The numbers, it so happens, are a sequence (1,3,6,10,15, . . .) that you can calculate with a simple formula called a generating function. In this case the generating function is
To get a number at any point in the sequence, like say the 5th spot, just plug in 5 for n
, and you get the answer
As it turns out, you can also use the generating function to figure out what the values would be between whole numbers. Essentially, you replace the number n
with an x
, which can have any numerical value you like. If you plot the result, you get this, for positive x
The curve you see here goes up to infinity as x
goes to infinity. So far, so good. But if we're going to plot the graph between the various values of n
, we might as well look at negative values of x
When you do that, you get a graph like this
There are three interesting areas in this graph. The area above the horizontal axis and to the right of the curve (let's call it A), the area above the horizontal axis and to the left of the curve (call it B), and the area trapped between the curve and the horizontal axis (which I call C).
A and B are large areas - infinite actually, as you extend the curves to infinity, but C is small. In fact, if you use calculus to determine it's size, it's 1/12.
And because it's below the axis, it's conventionally considered negative, so it's -1/12. It's an easy integral to do, but in case you're feeling lazy, I did it on Wolfram Alpha for you, just click here
Interesting, isn't it? The curve generated using the partial values of the series 1+2+3+4+ . . . gives you a graph with a little region in it that has an area of -1/12. Hmmmmm.
But you haven't seen anything yet . . . Consider the infinite areas A and B. If the curve were symmetrical around the vertical axis, the two areas would be exactly equal, provided you compare them out to the same distance to the right and left. That is, if the curve was centered on the graph (with the low point of the curve sitting on the y-axis), then the area of A from 0 to 10 would be the same as the area of B from 0 to -10. And this would stay true as you go out to infinity on the left and right. But our curve is pushed to the left, so the two areas are not the same any more. The area of A out to 10 is larger than the area of B out to -10. If you imagine flipping B to the right hand side, you can see the difference as the space between the curves.
(Update 2-5-14: I made a stupid error in this part, but I think the overall point is the same.)
In fact, it's easy to calculate the space between them out to any distance. Here's a Wolfram Alpha calculation of the difference that I performed by subtracting the area under B out to -100 from the area under A out to 100
. If you keep taking the area calculation to larger and large distances, the difference increases. (I ignored a tiny portion of the area, but it doesn't matter in the long run because both numbers are going to infinity eventually and a tiny missing piece makes no difference.)
The important point is that the space between these two curves (i.e. A-B) increases just like the sum 1+2+3+4+. . . (
It's not, actually, but it is infinite.)
That is, the difference between the infinity on the right and the infinity on the left is 1+2+3+4+ . . . (I checked this by subtracting the area between the curves from the partial sum of 1+2+3+4+ . . . , where the integrals and the series are evaluated out to increasing distances, to see how close they get. It's easy to see that they become increasingly close as you go to infinity. You can play with an example here.)
And of course, the difference between area A an B is itself infinity if you extend the curves out infinitely.
This is how 1+2+3+4+ . . . and -1/12 are associated. They aren't equal (update 2-6-14: provided you are taking limits), but -1/12, in the form of area C, is a characteristic of the curve While 1+2+3+4+ . . .
is the difference between the infinities on the left and right, which are also characteristic of the curve
. is the series that generates the curve in the first place.
So, despite my previous, non-mathematical argument to the contrary . . . Update 2-6-14: The limit as m goes to infinity of 1+2+3+4+ . . .+m does not equal -1/12.
(Update 2-6-14: but I still believe 1+2+3+4+ . . .= -1/12, if the ENTIRE infinity of whole numbers are included. This doesn't mean the series converges to -1/12. It doesn't converge, and nobody I know of ever said it does.)
As I see it, -1/12 is a kind of label for the curve that you can generate using partial sums of 1+2+3+4+ . . .
The same thing works for 1+2^3+3^3+4^3+ . . . , and 1+2^5+3^5+4^5+ . . . and so on for any odd power (i.e., zeta(-3), zeta (-5), etc). I used this method to calculate the associated values of the zeta function for powers up to 13. In each case, you get a specific value the area C that's associated with the zeta function that creates the curve.
Here's a list of the C areas I calculated for curves generated by several series
zeta(-1) = 1+2+3+4+ . . . ---> -1/12
zeta(-3) =1+2^3+3^3+4^3+ . . . ---> 1/120
zeta(-5) =1+2^5+^5+4^5+ . . . ---> -1/252
zeta(-7) =1+2^7+3^7+4^7+ . . . ---> 1/240
zeta(-9) =1+2^9+3^9+4^9+ . . . ---> -1/132
zeta(-11) =1+2^11+3^11+4^11+ . . . ---> 691/32760
zeta(-13) =1+2^13+3^13+4^13+ . . . ---> -1/12
They agree with the published values of the zeta function for negative integers
listed on Wikipedia.
(Although Wikipedia stops at zeta(-7) and I go to zeta(-13). The fact that the number associated with zeta(-1) and zeta(-13) are the same looks like potential trouble, BTW - after all, how would you know if your -1/12 is associated with zeta(-1) or zeta(-13)? It also suggests that the Numberphiles
could have shown that -1/12 = 1+2^13+3^13+4^13+ . . ., or that 1+2^13+3^13+4^13+ . . .= 1+2+3+4+ . . ., if they felt like it.)
This little procedure works for even powers too, except the answer is always zero. Here's what the C area looks for for the curve generated from zeta(-2)=1+2^2+3^2+4^2+ . . .
The parts above and below the axis cancel for all series of this type with even powers, and as a result the total area doesn't give you any information. As you can see for the integral of the curve that comes from1+2^2+3^2+4^2+ . . . How Could I Have Screwed Up So Badly?
So, I was wrong.
1+2+3+4+ . . . and -1/12 have a relationship with each other but certainly don't equal each other (Update 2-6-14: that is, the series doesn't converge to -1/12, but I still think it equals -1/12 if ALL the terms are included in the sum).
Now I understand, and I hope all you do to.
I'm going to blame my misadventure on the trouble with using words to describe mathematical ideas. Before working out this problem, there was no way I could understand what it means when someone says that the value of -1/12 "can be assigned to an infinite series." They sounded like gibberish,
but now I think I get it
Ramanujan is to blame a bit too. After all, how are we supposed to understand what he was trying to say here?
"I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal."-S. Ramanujan in a letter to G.H. Hardy
Are we supposed to realize that "under my theory" means that "=" doesn't mean equal?
I haven't found his original work, but several people have reproduced a calculation by Euler that uses an equal sign in the same way. If the two sides aren't equal then, as I recall from second grade math, you can't use an equal sign.
At least on person I spoke to said that in order to understand it, you have to know what the ". . ." in the expression 1+2+3+4+. . . means. As you can see from the example above, the dots mean exactly what they always mean. If they didn't, then we'd be in almost as much trouble as having equal signs that don't mean equal.
Finally, there are the physicists that say they need the relation 1+2+3+4+ . . . = -1/12. Some of them seem to believe the equation is what it is, and that our failure to understand it shouldn't stand in the way of using it. That struck me as the most romantic view, and it was the one I latched onto at the end of the day. Now, I realize that this is a far too mysterious view. It's interesting, but I hope this post shows
that it's really not that hard to understand
the connection between 1+2+3+4+ . . . and -1/12.
While 1+2+3+4+ . . . = -1/12 is clearly not true
(update 2-6-14: if you insist that the statement is taking a limit) I can see that using the relationship between them might give you a handy way to do things like subtract infinities in consistent ways. The number -1/12 is sort of like a leash that's attached to the infinities in the curve at the top of this post. As long as you hold on to the leash, and don't lose track of the infinities you're dealing with, then you can use them to tame some important physics problems - like those in QED and string theory. How that's done, exactly, is beyond me at this point. But I can sort of imagine what's going on.
I hope there are other, comparably bizarre mathematical controversies out there. One thing is for sure, though, I'm not going to rely on words, even from the most decorated experts in their fields, to try to understand things like this. I'm going to get out a pencil, fire up Wolfram Alpha, and just do the math.