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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

• 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 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.

• 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.

• 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.

• It's can't be.

• Oh.

• Hang on.

• Oh, that's actually quite convincing.

• It might be 1/2.

• 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

• 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'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.

• 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.

• 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.

• If Grandi's series is 1/2, what does that mean?

• Is it 1/2 on, 1/2 off?

• Is it on and off at the same time?

• What do you think?

• So he's given a head start.

• He's got 100 meter head start.

• And then they start the face.