Previously, I had written about the somewhat bizarre behavior of a divergent geometric infinite series, containing no negative numbers, but which appeared to add up to -1:
I discussed how this divergent infinite series actually represents the Taylor Series expansion of the analytic continuation on the complex plane of a function that does actually come out to equal -1 at a specific relevant point. (Both Taylor Series and analytic continuation are topics which warrant their own discussions; but, for now, we’ll hold off on that until another day.)
That article had also contained a video from Minute Physics which discussed this bizarre series, although without delving into details about why the series exhibited such bizarre behavior. Well, the fine folks from Numberphile have posted yet another video about a similarly bizarre series:
“You have to go to infinity, Brady.”
Now, it just so happens that this dovetails nicely into something I’ve been leading up to in my posts: the Riemann zeta function and the Riemann Hypothesis. I have a bit more groundwork to cover before diving into them; but, as it turns out, the Riemann zeta function comes into play in a secondary proof for the sum described in that video. That proof is covered in a secondary video:
It Doesn’t Add Up!
So here’s the bizarre series, the sum of all of the natural numbers:
Weird, huh? Obviously, the series doesn’t actually add up to this negative sum. It is a divergent series that blows up to infinity. A bit more on that in a moment. But first, here is the proof presented in the first Numberphile video.
First, let us take the following series, known as Grandi’s Series, an infinite series alternating between 1 and -1:
This series is a bit challenging to evaluate. Looking at the partial sums of an odd number of terms in this series will always yield a value of 1, whereas the partial sum of an even number of terms will always give 0. Now, it can be shown (and is shown in yet another Numberphile video), that the appropriate sum for this series is the average of the two values, 1/2.
Let us also consider the following series:
And our sum of interest is defined as:
First off, let us add S2 to itself:
Well, now we are getting somewhere. This yields
Now, let us subtract S2 from S:
Now, we’ve already figured out what S2 is, so we substitute that in and simplify:
So, there you have it.
Okay, how about a different, more rigorous proof? Let’s take a look at the proof from the second Numberphile video, a proof first discovered by Leonhard Euler. First of all, let us consider the following series:
It can readily be shown (but I’ll leave it as an exercise for the reader for now) that the above is strictly true for values of x less than one. Now let us differentiate with respect to x:
Now, let us set x = -1. What does this give us?
Now we bring the big guns to bear. Here is the Riemann zeta function:
Now, when Euler first worked with this function, he only studied in the context of s being a real number. However, Riemann extended his analyis of the function into the complex plane. We’ll focus for the time being on Euler’s view of the function, but keep in the back of your head that a more rigorous version of this requires considering the complex plane.
Now, let us do a bit of manipulation of the zeta function:
Next, Euler subtracted twice this expression from the original zeta function:
Now we set s=-1 and plug it in:
But we’ve already figured out above that this series is equal to 1/4, so:
Now, keep in mind that Euler was looking at this function on the real number line. The series in question is simply the special case of the zeta function at s=-1. However, there is a catch. In the realm of real numbers, the zeta function only converges for values of s larger than 1. What Riemann realized is that he could make the zeta function converge for all values of s, except for a pole at s=1, by switching to the complex plane and performing the analytic continuation of the function. In that context, the value of the zeta function at s=-1 is in fact -1/12.
No, Seriously, What’s Really Happening Here?
Okay, obviously, the series in question doesn’t REALLY add up to -1/12. That is an impossibility in terms of how we ordinarily define algebraic sums. The point here is that we are using a non-standard definition of sums involving analytic continuation on the complex plane. We aren’t just talking about the real number line. The “proofs” in these videos aren’t really rigorous and break a few rules. (Specifically, if you have a series that does not converge, you can’t go adding and subtracting other series to it willy-nilly. Cauchy taught us this.)
Since this distinction between different types of summation isn’t really made clear in the original video, this has caused a bit of a ruckus online. After I had already started writing this blog article, Phil Plait from the Bad Astronomy blog posted an article about the video, and a firestorm ensued in the comments and on Twitter. Mark Chu-Carroll over at the “Good Math, Bad Math” blog picked up on the topic, and Phil Plait subsequently posted a follow-up to clarify the situation.
So, what’s all of this about different types of sums?
Back in the middle of the 18th century, Leonhard Euler came up with a proof that the summation we are discussing does indeed equal -1/12. He also came up with the result that 1 – 2 + 3 – 4 + … = 1/4. Both of these results, although obtained rigorously as far as he could tell, seemed paradoxical to him. What was needed was a new way to deal with divergent sums. Towards the end of the 19th century, several mathematicians started coming up with those techniques, some of which were based upon Euler’s work.
The standard algebraic infinite sum is typically defined by taking the nth partial sum of the series in the limit as n goes to infinity. By and large, these alternative summation techniques involve differing definitions for the limits that are taken when evaluating an infinite sum, as well as modifying the domain over which the sum is taken (such switching to the complex plane). These summation techniques frequently focus on the properties of partial sums since they address scenarios where the complete sums are not well-defined. By using these techniques, we can assign meaningful values to series which do not converge to a finite value under standard algebraic summation.
For example, suppose we have an infinite series whose nth partial sum oscillates symmetrically as n goes to infinity, never converging on a specific value. An example of this would be the Grandi’s Series we mentioned earlier, whose partial sums alternate between 1 and 0. However, instead of using ordinary algebraic summation, we can define a summation in which we sum the series to the nth term, then divide by n, doing this for all values of n. If the series converges to a value in the limit as n goes to infinity, we say that the series is Cesàro summable, We’ve essentially taken the average of all partial sums in the series.
The Grandi’s Series can also be evaluated terms of something called an Abel sum. Similarly, 1 – 2 + 3 – 4 + … can be evaluated using Abel summation. And for our sum of all natural numbers, it turns out that their are two summation techniques which apply: zeta function regularization (which is a formalization of Euler’s approach) and Ramanujan summation.
But what does it mean when we say that we are finding a value for something that doesn’t have a value? Let us take a look at a generic geometric series:
Let’s multiply the whole thing by a factor r, and then subtract the original series from that:
For the original series to be convergent, r has to be less than one. In that scenario, evaluating this formula would give us a value equal to the limit of n partial sums of the series as the limit of n goes to infinity. For other values of r, where the series is divergent, we can still use this formula to get a value. This value is not the limit of n partial sums as n goes to infinity, since that limit does not exist. But we still have a value using the same formula. (For the special case of r=-1, we have an oscillating value, as with our Grandi’s Series.)
So, What Does This Have To Do With String Theory?
That would be a rather lengthy discussion, but have a look at a series of posts by mathematical physicist John Baez (who happens to be the cousin of singer Joan Baez) listed below in the “For More Information” section. He covers the topic quite nicely. This sum also shows up in other areas of physics, including QED calculations of the Casimir Effect.
For Further Information:
- The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation | What’s new
- Mathematical Physicist John Baez has a series of articles which hit upon this topic:
- The Reference Frame: Zeta-function regularization
- The Reference Frame: Why is the sum of integers equal to -1/12
- 1 + 2 + 3 + 4 + ⋯ – Wikipedia, the free encyclopedia
- 1 − 2 + 3 − 4 + · · · – Wikipedia, the free encyclopedia
- Grandi’s series – Wikipedia, the free encyclopedia
- Summation of Grandi’s series – Wikipedia, the free encyclopedia
- Zeta function regularization – Wikipedia, the free encyclopedia
- Ramanujan summation – Wikipedia, the free encyclopedia
- Cesàro summation – Wikipedia, the free encyclopedia
- Divergent series – Wikipedia, the free encyclopedia
- Divergent geometric series – Wikipedia, the free encyclopedia
- Analytic continuation – Wikipedia, the free encyclopedia