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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.