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  • 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.