In case you didn’t know, I’m a fan of the short youtube videos given by the user minutePhysics. The guy has a knack for exploring and explaining physics concepts in a simple way. But let’s cut to the chase.
In one of his videos, he makes the following claim :
You can see that video here. Of course, the internet does not forgive “errors” easily. Should we send minutePhysics to the mathematical jail or does he actually know his math (despite being a physicist :P)? Let’s see.
First things first, you don’t have to be a mathematician to see that grows without bounds. In modern terminology, we say that this sum converges to infinity. But then, how can it be -1? Can anyone really believe that this sum of positive numbers is summable to a negative number? If someone is suggesting this, then he/she must be crazy, made an error somewhere in his/her algebra or quite possibly both. But then again, something is nudging you in the back of your head. What gives?
Here’s a small mathematical story. Divergent series have been studied for a long time by many famous mathematicians, such as Cauchy, Ramanujan, Euler, Poisson and Fourier. Hardy, an English mathematician and collaborator of Ramanujan, wrote a whole book in 1940 about them and how to study them. The first thought that passes through your minds would be “what’s there to study in a divergent series?”.
Quite a lot actually.
Back then, mathematicians had already started to perfect in some sense a theory of convergent series. That is, they started to build a set of tools that could work in many cases and could tell them if a series converged to a finite number or not. And those tools ranged from the very basic, comparing for example the sum with the sum , to some that used sophisticated ideas from analysis, like integrals.
But science always progresses, so pretty soon divergent series came in focus again. Fourier started to study what we call today a trigonometric polynomial, with the hope of using them to approximate functions that he couldn’t compute explicitly. Here’s what’s a trigonometric polynomial. Pick real numbers and compute :
And that’s a trigonometric polynomial. For example, , are trigonometric polynomials. Here are their graphs.
You can probably tell which is which by looking at this graph. What is important here, is the next idea. What if, given a function f, you could find a pair of that depend on that f, such that you could approximate f by a trigonometric polynomial? By combining a number of ideas, Fourier found out that you can indeed pick such so that this idea would work and today we call that idea Fourier series and science students find it all over the place. A Fourier series is just the limit of the previous trigometric polynomial. In other words, pick an x and then compute :
and expect that this number will converge to the number
But back then, this idea was just beginning to develop and there were some problems that had to be addressed. And the first problem was that of convergence. And in many cases, you would pick a real number x, you would try to sum the numbers together, and the result wouldn’t have anything to do with what you expected. The series would just diverge to infinity or even not converge at all. I will talk about one way that was addresed in a while cause I want to show you another example, from even further back in time.
Luigi Guido Grandi was an Italian mathematician and priest. Around 1704 he was studying the peculiar sequence of numbers
which can also be generated as
So, curious as he was, he started to think about the infinite sum Of course, he didn’t have the calculus knowledge we have today but still, he could try stuff.
On first look, this sum shouldn’t exist, right? The argument is simple. Since the partial sums are “half” of the times on 0 and “half” of the times of 1, this sum does not converge and that’s that.
But let’s say that it could converge and let’s call that sum s. Then,
oops. It seems like by grouping the terms a bit differently, we arrived at an entirely different result. What happened? Pay attention cause this is a subtle point. 🙂
First of all, we did something different from before. We didn’t exactly try to directly sum the previous numbers, we know that this is impossible. What we did is that we regrouped those numbers with the hope of extracting a sligthly meaningful result about the previous series of numbers and we mean the equality in exactly that sense i.e that according to those manipulations that we did, the result is 1/2. This is not the strict mathematical equality. It’s equality in that sense. Do not think too much about this part yet, but do read on. 🙂
Here’s a different take on that example. Let’s say that we want to make some kind of numerical sense of the sum 1-1+1-1+1-1+…. From the previous argument, we know that just adding won’t cut it. The next term will always make us go from 0 to 1 or from 1 to 0. But say we need to assign some kind of number to that sum in a way that makes sense and tells us something more about the sum. A clever idea by Ernesto Cesaro is the following :
You want to compute for some numbers and let’s suppose that you know that this infinite sum exists and is a finite number. Then Cesaro says that you can compute and you will get the exact same number.
Proving that fact is a nice little exercise and perhaps you can intuitively see that as grows, will start to approach the infinite sum, so taking the mean value of those should give us back the same answer in more or less the same way as .
Now this is a very important property! It means that if we use this idea on a sum of infinite numbers, then if that sum exists and has a certain value in the usual way, then we will get exactly the same result by our idea.
What is interesting about the Cesaro summation is that we can apply it in cases where we don’t have the usual idea of convergence. If applied on the previous example, we have
where we added the first 2n partial sums of the Grandi sequence. This allowed us to find the sum, or in this case, the average of the sequence . It’s standard symbolism in mathematics to write this result as :
in the sense of Cesaro
by which we mean that if we apply to this sum the idea of Cesaro, then the value is 1/2. I repeat here that the previous statement does not denote equality between the sum and 1/2. It’s just a symbolism.
This idea of Cesaro has been applied to Fourier series as well and in other contexts. Furthermore, there is a whole class of such methods so that you can extract numerical values from divergent sums by summing them and doing a little something extra (which in the case of Cesaro was to divide by N).
So, to wrap it up, here are some questions I found :
– Can I make my own method?
You need to study some stuff first. I would recommend starting from this wikipedia entry. There are some things that you can do with a series and some things that you can’t do. Check the Riemman series theorem for a take on that.
– About MinutePhysics …
So, yeah. About him. When he wrote the equality for the infinite sum, he didn’t actually mean strict equality in the usual sense but in the same sense that we talked about in this post. He did a “small” manipulation of the series to extract a numerical value that tells something about this series, in the same way that a straightforward summation tells us that our sum is unbounded. It was legit. No numerical trick and no trolling. Well, perhaps a little bit of trolling, but he did explain it in his next video (if you were paying attention). 🙂
– But we are just adding numbers … are you saying that 1+1 is open to interpretation?
Not at all. We are adding an infinite amount of numbers and doing this addition in the usual way can only give us so much information about the sum. Using alternative methods of summing up, like the Cesaro idea, can give us a different kind of information. And remember that all those summation methods are consistent, meaning that they agree with each other when the usual method of just adding up the numbers gives meaningful results, i.e numbers.
– Last one. You said that by doing those “small” changes, you acquire meaningful information about the series that we can’t get in the usual way.
Right. So, in the example of , we said that our partial sums, what we get if we only add a finite number of terms, spend half of their time on 0 and half of their time on 1. The Cesaro sum then gave us as a result 1/2. Think about it.
– … But what about ? What does -1 mean for this series?
Sadly, I haven’t researched this enough to be able to answer this question. But if I were to do it, I would start from here.
I really hope this post will clear a couple of things. 🙂 I leave you with a quote by Hardy, “stolen” from his book on divergent series,
It does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition : it was not natural for them to say, in so many words, ‘by X we mean Y’. It is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define 1-1+1-1…?’ but ‘What is 1-1+1-1-…?’, and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.