Updated with a third footnote clarifying my use of the term “diverge,” thanks to suggestion by Evelyn Lamb, who has also written an excellent discussion of the problem with the video. At the end of this post I list all the critiques I’ve found so far.
I feel like one of those grizzled action heroes who, having given it all up, is dragged reluctantly out of retirement for one more big mission. Over the past month or so (honestly, I forget how long I was working on things), I wrote a series of blog posts on the “weirdness” of infinity in mathematical set theory. Hopefully, there were two things that I got across in those posts: (1) infinity can be very weird, but (2) it can be comprehended, and even reasonable, once one understands the assumptions and limitations built into the mathematics.
Having retired from writing those posts, the other day I came across the following video:
So, using a seemingly simple series of mathematical manipulations, they “prove” the following astounding result: the infinite sum of increasing positive integers equals a finite, fractional, negative number. In short:
This video was picked up by Phil Plait at Bad Astronomy, who called* it “simply the most astonishing math that you’ll ever see.” It has already spread far and wide across the internet, including making it to the popular site Boing Boing.
But is it true? The video makes it seem so simple, and uncontroversial, almost obvious. But there are some big mathematical assumptions hidden in their argument that, in my opinion, make it very misleading. To put it another way: in a restricted, specialized mathematical sense, one can assign the value -1/12 to the increasing positive sum. But in the usual sense of addition that most human beings would intuitively use, the result is nonsensical.
To me, this is an important distinction: a depressingly large portion of the population automatically assumes that mathematics is some nonintuitive, bizarre wizardry that only the super-intelligent can possibly fathom. Showing such a crazy result without qualification only reinforces that view, and in my opinion does a disservice to mathematics.
I’ve actually discussed this result years ago on this blog, talking about the Riemann zeta function and how -1/12 isn’t really equal to the infinite sum given. But even that discussion is probably a little too abstract, especially since I don’t discuss in any detail how the result -1/12 could be physically accurate. As it has been noted (and I’ve noted myself), the -1/12 result can be used with surprising accuracy in physics problems. But even there, things are much more subtle than they appear.
So let’s take a closer look** at the “proof” that an infinite increasing sum can equal -1/12. We will explain why the answer is not so simple as the video makes it appear, and why it is also not quite so simple to say that physics justifies the answer. We have a lot of ground to cover, so let’s go!
We should start by getting some definitions out of the way. An infinite sum of numbers is technically known as an infinite series. If we generally label these numbers as , , and so forth, the series S may be written as
Neglecting the “proof” above for the moment, it is rather intuitive that not every collection of numbers will sum to a finite value, or even a single value. We say a series converges only if, on adding more and more terms to the sum, we find that the total gets closer and closer to a single value. Some examples are worth pointing out. The following series,
converges to the value 2, for instance, while the series
diverges. For this latter series, as we add more and more of the terms to the total it just gets bigger and bigger, approaching the infinite.
Another series of note is the following:
This series does not diverge***, but it does not converge, either. As we add more terms to the total, it alternates between 1 and 0, and never fixes itself on either. We will call such a series oscillatory.
This brings us to the “proof” that . We can summarize the reasoning as follows (following the abbreviated description at Good Math, Bad Math):
Let’s assign a value 1/2 to the oscillatory series , i.e.
because reasons. Then, we consider the series , given by
and we add this series to itself, but with terms shifted slightly one place:
This suggests that , or that .
Now we introduce another series, , given as
and we subtract from :
What we have left over is simply 4 times , or we may (supposedly) write
which in turn implies
This seems deceptively simple, but there is an egregious oversimplification that really changes the entire meaning of this “proof,” and results in it not showing what it purports to show!
A little reflection shows that the entire argument hinges upon the statement that
In justifying this, the video just argues that summing an odd number of terms gives one, and an even number of terms gives zero. “Do we stop at an odd or an even point… well we don’t know, so we take the average of the two.” So simply stated, and so completely misleading!
There is a motivation to labeling the sum of the series as 1/2, and the sum of as -1/12, but the reasoning is much more complicated. For a moment, let’s put ourselves in the mind of a mathematician and see how they would view the problem.
We have seen that series may be described as convergent, divergent, or oscillatory, based on what happens as we increase the number of summed terms without bound. A convergent series sums to a well-defined number, but divergent and oscillatory series have no defined sums — they “blow up” or oscillate endlessly, respectively.
For brilliant mathematicians like Leonhard Euler (1707-1783), Niels Henrik Abel (1802-1829) and Srinivasa Ramanujan (1887-1920), however, this answer was unsatisfying. One way to think about it: there are many different divergent series, each with different terms, and they should be distinguishable in some manner. For instance,
both diverge, but they do so in different ways, with different terms in their respective series. A similar argument applies to oscillating series. It is not unreasonable to assume that there must be some way to quantify the behavior of even a nonconvergent series.
There are, in fact, several different ways to go about it. I will focus on one that strikes the best balance between simplicity and practicality, and which employs the geometric series, one of the few series whose sum can be found by a simple argument.
The geometric series is the series of the form:
where x is a variable whose allowable values are to be determined. What is the sum of this series, if it exists? Let us look at the finite partial sum of the series, defined as
For a finite N, we note the following relations:
These two expressions must be equal to one another, so we can solve for between them to get
Here’s the great thing: if we let , then we get the sum of the infinite geometric series, in a very simple form! Because of the term in the numerator, we quickly see that this will only approach a finite value if
If , the denominator of the sum “blows up”, and if , the numerator oscillates as a function of . If , then just gets bigger and bigger as increases. We may therefore say the following:
The geometric series converges for values of , and diverges for . For , the series is either divergent or oscillatory.
If we allow to be a complex number, i.e. , with , then we can say that the geometric series converges strictly inside a circle of radius 1 centered on the origin, and it is nonconvergent on or outside that circle.
If we could sum the geometric series for , we would see that it is exactly the series talked about earlier. However, our derivation of the geometric series sum demonstrates that this series has no sum!
This is where, as mathematicians, we decide to be tricky. Though has no sum, the geometric series has a sum for values of that are arbitrarily close to that also lie within the circle. We therefore decide to assign a value to the series , by saying, in essence:
The sum of the oscillating series is defined to be the sum of the geometric series that is “most like” but that still converges.
What is that geometric series that is “most like” ? Well, we start inside the circle and move as close as we can to the point without hitting it. As gets closer and closer to , the sum of the geometric series gets closer and closer to
which is the naive result claimed to be true in the video. This process is illustrated pictorially below.
Considering we are now 1600 words into this blog post, hopefully it is now clear that the argument is not as trivial as the video implies. In fact, to prove this result, we are not even summing the divergent series anymore: we are summing a (convergent) geometric series that is mathematically close to it.
The method just described is what is called Abel’s method, and the summation derived is called Abel’s sum. The crux of Abel’s method: we take our original divergent series, modify it so that it is as weakly converging as possible (“infinitely” weak, in a loose sense), and then assign the sum of this weakly converging series to our diverging series. In formulas, let us imagine that we have a diverging series given by
We create a new series , of the form
If , we have added a “damping” to the terms of the series, as each power of is smaller than the last. In the limit , we make this damping infinitely weak but, under the right conditions, the series may still converge.
This idea probably seems very abstract at this point. Let’s try and make an analogy to see what we’re actually doing! Imagine that the circle of convergence is actually a very high, very steep plateau, and that the dropoff begins at . Being the dreamers that we are, we would like to know what it is like to fly without using an airplane, but if we step off the plateau (enter the region ) we will fall to our death, because flying is impossible. What can we do? We instead inch ourselves as close as we can to the edge of the plateau without going off; when we reach that boundary, we define it as being “as close to flying as possible.”
Analogously, we can’t sum the geometric series for . We can, however, get as close as possible to that point and use the sum of the “closest” series as the defined sum of it.
Abel’s method works as well for our series , which was defined as
The corresponding Abel series is
which can again be shown to be convergent within the circle ; its sum within this circle can be shown to be
For , we arrive at the series . If we again get as close as we can to without reaching it, we find we can assign a sum to the diverging series of the form
which is in agreement with the “proof” result. Again, though, it is important to note that we are not summing the original series: we are summing an infinitely close but distinct series that has a finite sum.
It is also possible to determine an Abel-like sum for the series , and show that that sum is , but it is actually more complicated: the simple Abel argument we have used here is insufficient to “fix” the series , and more general methods must be used. The conclusion is the same, though: we are not summing the series in the usual sense of the term. If you were to truly sum the series , all the way out to infinity, you would not get -1/12.
Two questions remain. First: why, then, does the “proof” given in the video give the right answer? My suspicion is that it works because the proof is implicitly summing the Abel-like series, without having the explicit values of present! It is a calculation involving convergent series disguised as one involving divergent series.
The second question: the series sum
does in fact work in theoretical calculations, and does produce results consistent with experiment: doesn’t that mean that the sum is in fact correct? Here I make a point similar to one that Blake makes at Science After Sunclipse:
Suppose you have done some sort of complicated physics calculation and the result comes out to directly involve a divergent series. A good example of this is the Casimir effect, in which quantum fluctuations of the vacuum provide an attractive force between a pair of closely-spaced metal plates.
In fact, a calculation of this force directly depends on the series ! One then has two choices: assume that the calculation is completely wrong, as the force is demonstrably not infinite, or assume that there is some weak “damping” effect of unknown nature that brings the system into line. Including this “damping” is analogous to constructing an Abel-type series: we introduce an extremely weak decay into our series that makes it converge.
The wonderful thing is that the nature of this damping does not strongly effect the result! Any appropriate weak damping will push the Casimir series to the finite sum -1/12, just like the construction of the Abel sum of the series pushed the sum to approximately 1/2. The use of a finite value as the sum of a divergent series is not a supernatural mathematical phenomenon, but an acknowledgement that other unknown physics is at work.
We’ve done a lot here, so let me try and summarize. First: the “series” does not have any finite sum in the normal sense of the word “sum.” Second: there are ways, however, to assign a finite “sum” to such divergent series in a mathematically consistent way, though we stress again that this is not a “sum” in the usual sense of the word. Third: The generalized sum of a divergent series has use in physics problems, as a way to incorporate unknown damping factors into a problem that seemingly has a divergent solution. Fourth: the original video does not note any of these points.
So why is this important? Part of what I’ve tried to show on this blog is that mathematics and physics can be extremely non-intuitive, even bizarre, but that they have their own rules and logic that make perfect sense once you get familiar with them. The original video, in my opinion, acts more like a magic trick than an explanation: it shows a peculiar, non-intuitive result and tries to pass it off as absolute truth without qualification. Making science and math look like incomprehensible magic does not do any favors for the scientists who study it nor for the public who would like to understand it.
* Phil has since added a post explaining that there is more to the math than the video indicates.
*** I should note that mathematicians usually say that any series that does not “converge” must “diverge” by definition, including both series that “blow up” and “oscillate.” I use a looser terminology where “diverge” specifically means “blows up.”
List of all the critiques I’ve found so far, for those who would like to get other perspectives: