[Update: fixed an error that, while embarrassing, has no real bearing on anything, except perhaps to reinforce the admonition to be careful…]
The internet has been doing a whole lot of hating on this Numberphile video demonstrating (or, if you’re one of the haters, pretending to demonstrate through rank chicanery) that . I first saw it via Phil Plait of Slate’s Bad Astronomy blog, who followed his initial mind-blown and gushing post with a rather abject retraction. Some bloggers seem to be taking it quite personally, perhaps outraged that the charlatan Numberphilics are sullying the august and ethereal beauty of mathametics, ruining it for the rest of us and deluding the ignorant masses. Or something. I don’t get it, maybe because the original video is, if awfully (and admittedly) loose and sloppy and glib, pretty much correct.
Much of the vitriol is in the form of complaints that the manipulations in the video are not valid, that most manipulations with infinite series are not in fact valid, that one has to be very careful with infinite series, and that it’s easy to prove anything you want with innocuous-looking sleight of hand. That’s true: it is very easy to get these things wrong. I agree that Numberphile could have been more emphatic in pointing this out. But everything they do is legit. It’s a near thing — what they do in one context does not work in another — but it’s not by accident or legerdemain that their demonstration gives the right answer.
The first thing to get out of the way is that no, of course the sum doesn’t add up to -1/12 in the conventional way that you learned in elementary school. No infinite series has a sum in quite the way that 1+2=3. When we talk about sums of inifinite series what we really mean is that we have a summation method, a procedure that takes a series and produces a number, the “sum,” and that the procedure has some set of nice properties. Cauchy summation, taking limits of partial sums, is a particularly nice summation method with particularly nice properties, so nice that we have a special name — “convergent” — for the series for which it works. But it’s worth remembering that taking limits is already a non-trivial and potentially subtle operation.
In practice, what it means to say that is that if you find yourself adding up all the positive integers and expecting to get an answer, that answer is very likely to be . (Or that you’ve made a mistake somewhere, always a possibility.) That’s certainly the case in string theory, where one common treatment is to hit the infinite sum, point out that it’s hard, and then show from other constraints on the theory that the only workable answer is . Yes, we could and arguably should make up some other notation, but as far as I know there’s nothing standard, and we (especially physicists) abuse notation all the time. It is not nonsense, and it really is deep. And weird.
The usual way to justify the result, and the one that most of the internet thinks Numberphile should have used, is through the Riemann zeta function. In fact, Numberphile did use that in their follow-on video and in Tony Padilla’s response to the whole kerfuffle. That’s fine and correct and all, but (IMO) not a very intuitive way of looking at it. What do zeta functions and analytic continuations have to do with this? Lots, actually, but I’m dense and have always found it to be rather opaque. But there are other ways of looking at the problem, to my taste much more satisfying. One fortuitous consequence of this brouhaha has been a lot of links to a post of Terry Tao’s of a few years ago, which I have found enormously helpful. It’s pretty dense, especially if you’re not a mathematician, but worth reading even if you finding yourself glazing over at Bernoulli numbers. It is my chief inspiration in what follows (with the usual caveat the everything I have gotten wrong herein is my fault alone).
Beginning to hint at formality, let’s look at summation methods and their properties. Then we can prove things of the form “if this summation method has these properties, and it works for these series, then it must assign this series this value.” That’s one way to interpret the video sensibly. The problem is that different summation methods allow different manipulations, and figuring out which you can use is subtle. The arguments in the video (if you interpret it as I do) require three different methods, which get progressively more restrictive and finicky about what techniques you can legitimately use. What each of the video’s steps relies on does not work at the next step!
So let’s talk about summation methods and their properties.
Properties first, mostly stolen from wikipedia:
- Regularity: the method gives the ordinary sum for convergent series.
- Linearity: the methods behaves as you would expect when you add series together or multiply them by constants.
- Stability: You can offset a series and get the same answer : . Not all methods, including ones we’ll use, are stable in this sense!
- [no name that I know of!] We can “expand” the series by sticking zeros before every term and get the same answer:
This is not the same as inserting zeros in arbitrary places, including after each term — we can (and will) have series for which !
And some summation methods
- Cesàro summation: even if the partial sums don’t converge, their averages might, and if they do call that the sum.
- Abel summation: if converges for , and if exists, call that the sum.
- “Polynomial renormalization”: I don’t know what to call it, but what Terry Tao does in his deservedly much-quoted blog post mentioned above. Basically, we first “regularize” the sum by applying a smooth cutoff function that is 1 at 0 and 0 for . If the result is a polynomial in N plus some residual term that goes to 0 as N goes to infinity, we use just the constant part of the polynomial as the sum. Not coincidentally, this is very reminiscent to what physicists often do in quantum field theory.
Cesàro summation and Abel summation have all the properties I listed; polynomial renormalization, however, is not stable. Any series that is Cesàro summable is also Abel summable (and both give the same answer). I think that Abel summability implies and is consistent with polynomial renormalization (there is probably a proof lurking in the appropriate section of Tao’s article, but I haven’t figured out whether it works here). The series here, however, are all PR-summable, consistently with Abel summation for the ones that are Abel summable.
And now we’re ready to go through Numberphile’s argument in detail. We start with “Grandi’s series”
This is obviously divergent — the partial sums oscillate between 1 and 0 — but it Cesàro summable, as the partials averge out to .
Next we look at
This is Abel summable, and we could easily use the explicit formula to calculate its value, but instead we’ll just assume it and calculate indirectly, using linearity and stability and the value we already have for Grandi’s series:
so and .
Now things get subtle. Let’s assume that this carries through to PR summability (it does), and continue. We can no longer use stability — no more offsetting a series — but we can use the expansion-by-zeros trick if we do it right. We compute
from which we find , or .
To recap, we
- used Cesàro summation to compute a “sum” for Grandi’s series
- assumed that is Abel summable, and computed its value using Grandi’s series and the properties of Abel summation, and finally
- used that value and the properties of PR summation to compute the original sum.
As promised steps two and three were of the form “if this method works, the result has to be that.” It turns out that they do!
It’s worth looking at some manipulations that superficially appear to be every bit as valid as what we’ve just done but don’t work. First, I’ll state without proof that PR summation leads to the result that
If we subtract this from , we get
— I warned you offset wouldn’t work here. I also warned you that you couldn’t “expand” a series with zeros in the wrong place, and in fact we can calculate that
Go figure! Some others:
So it behooves you to be careful.