(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.)
For a longer explanation, read on . . .
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
for n=1 the partial sum is 1
for n=2 the partial sum is 1+2 = 3
for n=3 the partial sum is 1+2+3=6
for n=4 the partial sum is 1+2+3+4=10
for n=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
for n=5, G(5)=5(5+1)/2=15
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 too.
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.
(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.)
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+ . . .
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.
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?
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,
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
While 1+2+3+4+ . . . = -1/12 is clearly not true
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.