A Numberphile video posted earlier this month claims that the sum of all the positive integers is -1/12.

I'm usually a fan of the Numberphile crew, who do a great job making mathematics exciting and accessible, but this video disappointed me. There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series. Furthermore, the way it is presented contributes to a misconception I often come across as a math educator that mathematicians are arbitrarily changing the rules for no apparent reason, and students have no hope of knowing what is and isn't allowed in a given situation. In a post about this video, physicist Dr. Skyskull says, "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."

Roughly speaking, we say that the sum of an infinite series is a number L if, as we add more and more terms, we get closer and closer to the number L. If L is finite, we call the series convergent. One example of a convergent series is 1/2+1/4+1/8+1/16…. This series converges to the number 1. It's pretty easy to see why: after the first term, we're halfway to 1. After the second term, we're half of the remaining distance to 1, and so on.

Zeno's paradox says that we'll never actually get to 1, but from a limit point of view, we can get as close as we want. That is the definition of "sum" that mathematicians usually mean when they talk about infinite series, and it basically agrees with our intuitive definition of the words "sum" and "equal."

But not every series is convergent in this sense (we call non-convergent series divergent). Some, like 1-1+1-1…, might bounce around between different values as we keep adding more terms, and some, like 1+2+3+4... might get arbitrarily large. It's pretty clear, then, that using the limit definition of convergence for a series, the sum 1+2+3… does not converge. If I said, "I think the limit of this series is some finite number L," I could easily figure out how many terms to add to get as far above the number L as I wanted.

There are meaningful ways to associate the number -1/12 to the series 1+2+3…, but I prefer not to call -1/12 the "sum" of the positive integers. One way to tackle the problem is with the idea of analytic continuation in complex analysis.

Let's say you have a function f(z) that is defined somewhere in the complex plane. We'll call the domain where the function is defined U. You might figure out a way to construct another function F(z) that is defined in a larger region such that f(z)=F(z) whenever z is in U. So the new function F(z) agrees with the original function f(z) everywhere f(z) is defined, and it's defined at some points outside the domain of f(z). The function F(z) is called the analytic continuation of f(z). ("The" is the appropriate article to use because the analytic continuation of a function is unique.)

Analytic continuation is useful because complex functions are often defined as infinite series involving the variable z. However, most infinite series only converge for some values of z, and it would be nice if we could get functions to be defined in more places. The analytic continuation of a function can define values for a function outside of the area where its infinite series definition converges. We can say 1+2+3...=-1/12 by retrofitting the analytic continuation of a function to its original infinite series definition, a move that should come with a Lucille Bluth-style wink. Analytic continuation Lucille says, "I'm going to stick an equals sign between the value of the analytic continuation of a function at a point and the infinite series that defines the function elsewhere." Video from Fox, gif from fanpop.com.

The function in question is the Riemann zeta function, which is famous for its deep connections to questions about the distribution of prime numbers. When the real part of s is greater than 1, the Riemann zeta function ζ(s) is defined to be Σ∞n=1n-s. (We usually use the letter z for the variable in a complex function. In this case, we use s in deference to Riemann, who defined the zeta function in an 1859 paper [pdf].) This infinite series doesn't converge when s=-1, but you can see that when we put in s=-1, we get 1+2+3…. The Riemann zeta function is the analytic continuation of this function to the whole complex plane minus the point s=1. When s=-1, ζ(s)=-1/12. By sticking an equals sign between ζ(-1) and the formal infinite series that defines the function in some other parts of the complex plane, we get the statement that 1+2+3...=-1/12. Divergent series Lucille says, "1+2+3...=-1/12." Video from Fox, gif from fanpop.com.

Analytic continuation is not the only way to associate the number -1/12 to the series 1+2+3.... For a very good, in-depth explanation of a way that doesn't require complex analysis—complete with homework exercises—check out Terry Tao's post on the subject.

The Numberphile video bothered me because they had the opportunity to talk about what it means to assign a value to an infinite series and explain different ways of doing this. If you already know a little bit about the subject, you can watch the video and a longer related video about the topic and catch tidbits of what's really going on. But the video's "wow" factor comes from the fact that it makes no sense for a bunch of positive numbers to sum up to a negative number if the audience assumes that "sum" means what they think it means.