In the previous post, we looked at sums of the first *n* consecutive powers, i.e. sums of the form

where *p* was a positive integer. Here we look at what happens when we let *p* be a *negative* integer and we let *n* go to infinity. We’ll learn more about Bernoulli numbers and we’ll see what is meant by apparently absurd statements such as 1 + 2 + 3 + … = -1/12.

If *p* < -1, then the limit as *n* goes to infinity of *S*_{p}(*n*) is ζ(*-p*). That is, for *s* > 1, the Riemann zeta function ζ(*s*) is defined by

We don’t have to limit ourselves to real numbers *s* > 1; the definition holds for complex numbers *s* with real part greater than 1. That’ll be important below.

When *s* is a positive even number, there’s a formula for ζ(*s*) in terms of the Bernoulli numbers:

The best-known special case of this formula is that

1 + 1/4 + 1/9 + 1/16 + … = π^{2} / 6.

It’s a famous open problem to find a closed-form expression for ζ(3) or any other odd argument.

The formula relating the zeta function and Bernoulli tells us a couple things about the Bernoulli numbers. First, for *n* ≥ 1 the Bernoulli numbers with index 2*n* alternate sign. Second, by looking at the sum defining ζ(2*n*) we can see that it is approximately 1 for large *n*. This tells us that for large *n*, |*B*_{2n}| is approximately (2*n*)! / 2^{2n-1} π^{2n}.

We said above that the sum defining the Riemann zeta function is valid for complex numbers *s* with real part greater than 1. There is a unique analytic extension of the zeta function to the rest of the complex plane, except at *s* = 1. The zeta function is defined, for example, at negative integers, but the sum defining zeta in the half-plane Re(*s*) > 1 is NOT valid.

You may have seen the equation

1 + 2 + 3 + … = -1/12.

This is an abuse of notation. The sum on the left clearly diverges to infinity. But if the sum defining ζ(*s*) for Re(*s*) > 1 were valid for *s* = -1 (which it is not) then the left side would equal ζ(-1). The analytic continuation of ζ *is* valid at -1, and in fact ζ(-1) = -1/12. So the equation above is true if you interpret the left side, not as an ordinary sum, but as a way of writing ζ(-1). The same approach could be used to make sense of similar equations such as

1^{2} + 2^{2} + 3^{2} + … = 0

and

1^{3} + 2^{3} + 3^{3} + … = 1/120.

I think it is really a disservice to non-mathematicians to be presenting the sum of the integers equal -1/12 “result” because the context to understand it is specialized and esoteric and just is not accessible to the layman. Tube channels like Numberphile shouldn’t try and I can’t help but think the motives behind such context-dropping is bad, i.e., to impress and/or intimidate others or to destroy math (nihilism). (To be clear, that doesn’t apply to this blog 🙂