>Two years ago in post #333, I linked to a Numberphile video that showed that if you list all whole numbers from one to infinity and sum them, the answer will be -1/12, which I still do not understand.That's understandable. That Numberphile video (
http://www.numberphile.com/videos/analytical_continuation1.html) really rubs me the wrong way; it's way too proud of its own cleverness and doesn't take the time to explain what's actually going on. And what's actually going on is pretty straightforward once you understand where the trick is.
So, allow me as we jump down the rabbit hole. Maybe grab a glass of water. Or your favorite adult beverage.
The video starts with 1 + 2 + 3 + 4 + ..., an infinite series. Dude #1 asks dude #2 what he thinks the sum is, and dude #2 says, "I think it would tend to infinity." Dude #1 says, "Actually, it's -1/12."
Here's the thing: they're both right. (And dude #2 is slightly more right.)
Under the basic rules of math, it's impossible to sum an infinite series. They're infinite. You can't add them up. You'd never finish; there's always more to be done.
At best, you can see what the partial sums approach, if anything. Let's start with a nicer one: 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
| 1 + 1/2 = 1.5 | 1/2 short of 2 |
| 1 + 1/2 + 1/4 = 1.75 | 1/4 short of 2 |
| 1 + 1/2 + 1/4 + 1/8 = 1.875 | 1/8 short of 2 |
| 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375 | 1/16 short of 2 |
This tends towards 2. We defend that with a game of HORSE. You give me a number anywhere just short of 2, and I can find a string of fractions there that gets closer to 2 than the number you chose. 1.9999? No problem: keep going until you get to the term that's 1/16,384, and you'll sum to around 1.999939.
On the other hand, the partial sums of 1 + 2 + 3 + 4 + ... are 1, 3, 6, 10, 15, 21, 28, ... These numbers increase without bound; they don't tend toward any number and they are always getting bigger. So we say it tends toward infinity. Just like dude #2 said.
There are some really interesting series out there that might be fun to play with, like the following oscillating series:
| 1 + -1 + 1 + -1 + 1 + -1 + 1 + -1 ... |
The partial sums of this one get weird:
| 1 = 1 | = p(1) |
| 1 + -1 = 0 | = p(2) |
| 1 + -1 + 1 = 1 | = p(3) |
| 1 + -1 + 1 + -1 = 0 | = p(4) |
| 1 + -1 + 1 + -1 + 1 = 1 | = p(5) |
The partial sums don't tend towards anything. Instead, they alternate between 1 and 0. Neither one can stand alone as the actual partial sum.
"But PFLATS, I can pair them up and add that way! It's clearly 0! Look!"
| (1 + -1) + (1 + -1) + (1 + -1) + ... = 0 + 0 + 0 + ... = 0. |
To which I say, awesome insight. But I can do the same thing to get 1 instead:
| 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1. |
So, as it stands, just like 1 + 2 + 3 + 4 + ..., we can't say that the partial sum tends towards any number. We call it divergent.
Mathematicians hate not being able to do something. You might remember the number i, the square root of -1, which was invented because math people didn't like not being able to take the square root of negative numbers. i turned out to be really useful, so it got to hang around.
In "normal" math, you can't take the square root of -9. It doesn't work: positive times positive is positive. Negative times negative is positive. Zero times zero is zero. Every number has to fall into one of those three categories, so there is no square root of -9. No number times itself can be negative. Until you extend the number system to allow for this new idea called imaginary and complex numbers, then you can get an answer: 3i.
Anyway, same thing here. In this case, math people said, "Okay, let's say you could actually add up all these numbers. How should that work, and what would happen?"
Well, it turns out, whenever you decide to say, "Okay, let's say you can do [impossible math thing]", you're making a deal with the math devil. To be able to do something forbidden in math, you have to give something up in exchange.
With the square root of negative 9, you gave up the ability to put your numbers in a strict order. Given -5, 3, -1, 0, and 12, you can order them: -5, -1, 0, 3, 12. When you introduce complex numbers into the mix, you lose that ability. Given -5, 3 + -4i, 3, -3 + 4i, 5, 3i, and 2 + 2i, you'll be hard pressed to put them in a good order. Which is bigger: 3 or 3i? 3, 3i, or 2 + 2i? 3 + -4i or -3 + 4i? Is -5 bigger or smaller than those? [1]
Anyway, say we decide, "You know what, I want to be able to sum every finite term of 1 + -1 + 1 + -1 + ... ! Hey, math devil, c'mere!"
The math devil will tell you to use a technique called analytic continuation[2]. With that in hand, we say that since our partial sums alternate between 1 and 0, we might as well call the overall sum one half, 0.5.
The video blows by this, but this is a
monumental choice. Think about it. How on earth should 1 + -1 + 1 + -1 + ...
EVER be 0.5? That number doesn't even show up!
Well, we've chosen it, and it works. It even works out surprisingly nicely if you dig into it[3]. It has some really useful properties, especially in physics.
But to work around infinity, we made a deal with the math devil. And those don't always work out well. It's mildly weird that we are living in a math world where 1 + -1 + 1 + -1 + ... comes out to a decimal. But we chose to enter it.
Now that we're living in that world, we can sum other divergent infinite series, which we
know should be impossible. Well, now it's not. But I'm not sure you'll like the results.
Pick up the video at that "sum is 1/2" part:
https://www.youtube.com/watch?v=w-I6XTVZXww&t=160The fact that 1 + 2 + 3 + 4 + ... = -1/12 is
a direct consequence of our decision to (use analytic continuation to) let 1 + -1 + 1 + -1 + ... = 1/2.
And in both cases, those
are the best answers we can get. 1/2 is the value you get when you dig into 1 + -1 + 1 + -1 + ... . If 1 + 2 + 3 + 4 + ... does have a sum, it'll have to be -1/12.
[1] -5, 3 + -4i, -3 + 4i, and 5 are all the biggest, and all the same "size", 5, because they're all 5 units away from 0. 3i and 3 come next, because they're both 3 units from 0. 2 + 2i is sqrt(8) = 2.828 units from 0. Represent each one as a point, like (-3, 4) for -3 + 4i, and calculate the distance from (0,0).
[2] Okay, analytic continuation. This is the mathy stuff. Don't worry, the next endnote is lighter.
We're not actually adding all the terms of the series. We're using the sum of
analytic continuation of the series, which is an imposter. If we look at our partial sums back in the table, we have p(1), p(2), and so forth. Partial sum 1, partial sum 2, partial sum 3. p(1.5), the partial sum of the first 1.5 terms, doesn't make any sense in that context.
What analytic continuation does (*WAVES HANDS WILDLY*) is to find an imposter equation that can replace p(1), p(2), etc., but can also handle p(1.5) and lots of other values. We'll call it big P. All P needs to do is give the right answer for P(1), P(2), and so on, while not doing anything too ridiculous in between those important numbers. Since P(1) = p(1), P(2) = p(2), and so on, P can replace p everywhere it matters without any major side effects.
And it turns out there's only ever one imposter equation that will work, which is called the analytic continuation of p. And we can use advanced calculus to sum P. It comes out to 1/2.
[3] In particular, say we decide to call its sum S. Then S = 1 + -1 + 1 + -1 + ... . We can note that 1 - S = 1 - (1 + -1 + 1 + -1 + ...). Distributing the negative sign gives 1 - S = 1 - 1 - (-1) - 1 - (-1) - ... . Fixing that to remove double negatives and only use addition gives 1 - S = 1 + -1 + 1 + -1, which... is just S! So 1 - S = S, 1 = 2S, and S = 1/2.
