字幕表 動画を再生する 英語字幕をプリント So today I want to try and bend your brains a little bit today. And I'm hoping it will cause a little bit of debate on the comments, because I know YouTube's the home of rational and informed debate. So I look forward to that. The question is what is this equal to? It's quite a simple sum. It starts with 1. Then I'm going to subtract 1. Then I'm going to add 1 again, then subtract 1, then add 1, then subtract 1, then add 1, then subtract 1. And I'm going to do this forever. You get the idea of that, I hope. So what does that equal to? So one of the answers that it might be is if I put the brackets like this-- here and here and here and here-- you can see each bracket is 1 minus 1 plus 1 minus 1 plus 1 minus 1. Each bracket is 0. So you're getting 0 plus 0 plus 0 plus 0 forever. So that's going to be equal to 0, isn't it? That's one of the answers it could be. The problem is there is another answer. If I do it again, we could put the brackets here, like this. Now let's say this is-- plus again there plus this bracket. So I started with 1 plus minus 1 plus 1-- that's a 0-- plus minus 1 plus 1. That's a 0. Et cetera, et cetera. All the brackets are 0. So all the brackets add up to 0. But I've got a 1 at the start. So now this is equal to 1. I've got two answers. I've got 0 if I put the brackets here. I've got 1 if I put brackets in a different place. There is a third answer as well, and this is the very weird one. Let's say it has a number, so let's call it S. We're going to try and find out what S is equal to. That's what we want to know. Let's do 1 minus S. So it's 1 minus this infinite sum. Let's do that. So let's write it out. Plus 1 minus 1 plus 1 minus 1-- right. If we take the bracket away, this minus number will mean that all the signs will get flipped, so you'll get 1 minus 1 plus 1 minus 1 plus 1 minus 1. That's what happens when I take away the bracket. But what I end up with is the thing I started with. That's just the alternating plus and minus 1. So I've got S again. So I've got 1 minus S is equal to S. That's OK. That's fine. You can solve that. In other words, if I take the S to the other side, I've got 2S equals 1, which then you can see that S is equal to 1/2. That's a weird answer. I've got a 1/2. The sum of adding plus and minus 1 forever give you a 1/2. Well, it might be 1. It might be 0. But it might be a 1/2. So the guy who came up with this idea was an Italian mathematician called Grandi. He did this in 1703. He was a monk. He was a mathematician. He was one of those types. And he published this. And he said this is weird. It's 0. It's 1. It's 1/2. What's that all about? And the mathematical community had a look at it. And they said well, it can't be 1/2, can it? I mean, you've got 1s and 0s. That's madness. It's can't be. Oh. Hang on. Oh, that's actually quite convincing. It might be 1/2. So there was a debate about this for a long time-- I think 150 years-- quite a debate until the 19th century, when all this stuff with infinite sums really got sorted out. A lot of people think that the best answer is 1/2. I want to try and show you why they think the best answer is 1/2. And then the one after that, I'm going to show you one more thing to completely bend your brain. If we pick a nice infinite sum-- because there are nice infinite sums, and there are bad infinite sums-- one of the nice ones is this. 1 plus 1/2 plus 1/4 plus 1/8 plus 1/16. And the way you can work out the answer for that-- actually I'm going to show you the proper way to do it. The proper way to do is look at the partial sums. We're going to add this sum term by term. So let's just make a sequence of them. I start with 1. I'll write that down. What do I get if I add the first two terms? It's 1 plus 1/2. It's actually 3/2. If you prefer, that's 1.5. Let's add the first three numbers together. So 1/2 plus 1/4. Let's do that. 7/4-- it's 1.75. If I add the first four together-- 15 over 8, which is 1.875. And if you did the next lot, you get 63 over 31-- 1.96875. You might be able to see, they're getting closer and closer to the value 2. In general, if I picked one in general, it would be 2 minus 1 over n. And if you can see, as the n gets bigger, this gets tiny and disappears, and you're just left with 2. And mathematicians are justified in saying that the whole infinite sum is equal to 2. If we try with Grandi's series, it doesn't work. Look at the partial sums. The first one is 1. And you add the first two together, you're going to get 0. You add the first three together, you get 1 again. You add the first four together, you get back to 0. And it keeps alternating between 1s and 0s. And it's not getting closer to a value. So this doesn't work with Grandi's series. So I'm going to show you a second method to work out sums. I'm going to take the partial sums, and I'm going to look at the averages. I'm just going to average as I go along. Almost the same way. I'll do it with this one first to show you the idea. Let's take the first one. That's 1. I'm going to add the first two partial sums together. So 1 plus 1.5, but I'll average it. I'm going to divide by 2. So it's going to be 1 plus 3/2 and then average it like that. Average is actually equal to 5/4. If I took the first three and averaged them, I would have 1 plus 3/2 plus 7/4 divide by 3. And that gives me 17 over 12, and-- well, hopefully, you get the idea of that. Again, the numbers are tending closer and closer to 2. It's just another method to get the same answer. It gives me 2 again. In fact, in general, what you get is 2 minus some junk. Oh, the joke isn't important. Look. It's junk. But this junk is getting smaller and smaller and smaller. So you're getting 2 again. It's just another way to find the same answer. But this method can be used with Grandi's series. Let's try it. We're averaging the partial sums. So those are the partial sums. We start with 1. Then if you average the first two, you get 1 plus 0 divided by 2, which is 1/2. Take the first three, and then divide by 3 gives me 2/3. I take the first four-- 1 plus 0 plus 1 plus 0-- divide by 4. That's another 1/2 again, if I get that right. Take the first five-- so you might be able to see what's going on, yeah-- divide by 5. So that's 3/5. What happens is, in general, you keep going. In general, you'll get 1/2 followed by something like 1/2 plus 1 over 2n. There we go. Again, and so you get some junk here that's going smaller and smaller and smaller. This is all tending towards 1/2. So together you're zoning in onto the number 1/2. So this is more technical than the other version I did but it's a second way to get sums. You average the partial sums. But it works for Grandi's series. It gives me 1/2. So what's going on? What's the difference? This second method-- it gives you sums when there are sums to find. A limit is when you're getting closer and closer to the value. Now Grandi's series does not have a limit, because you're not getting closer and closer to the value. But you have this second way of finding a sum. It's almost like a limit, but it's not really a limit. It's a fake limit. It's a pseudo limit. It has all the properties of limits. It does all the same things. It's so close to being a limit, that it turns up in calculations where you expect limits to turn up. But the difference is you're not getting closer and closer and closer. To really bend your brain, try and imagine this. We're going to try to imagine doing this in the real world. Imagine a light. We're going to turn the light on and off. So you turn the light on. You turn the light off. Now every time, if I go along Grandi's series, every time I see a 1, I turn the light on. Every time I see a minus 1, I turn the light off. So you turn it on, you turn it off. You turn it on, you turn it off. The partial sums actually tell you if the light is on or off. If you have a 1, that means you just turned in on. If you have a 0, that means you've just turned it off. You're going to start an experiment. After one minute, you turn the light on. After half a minute, you then turn the light off again. After a quarter of a minute, you turn the light on. After an 1/8 of a minute, you turn the light off. And you're turning it on and off, but you're doing it quicker and quicker and quicker. So you're doing that infinitely many times. But if we add up the time together, 1 minute plus 1/2 minute plus 1/4 of a minute plus 1/8 of a minute-- forever-- adds up to 2 minutes. In fact, that's that series I did there. If you remember the video we did about Zero's Paradox, that's not just getting closer to two minutes, you can actually compete it and finish the whole process in two minutes. So in two minutes time, you'll have turned on-- on and off-- the light infinitely many times and completed it. After two minutes, is the light on or off? If Grandi's series is 0, that means the light is off. If Grandi's series is 1, that means the light is on.