Tools of the Trade (for infinite sums) - Numberphile

0:00 Well, I'm going to tell you about um

0:02 a formula that's uh emerged that we've solved that's part

0:06 of a calculation that we're doing to calculate how many

0:10 primordial black holes could have formed in the early universe.

0:13 Primordial black holes formed almost as soon as the universe began in in in many

0:18 models and they could have formed the dark matter of the universe,

0:22 but we don't know how many there were and what masses

0:25 they had and how many of a given mass there were.

0:28 And so that's a calculation people try and do and I'm trying to do it.

0:32 But as part of it, this sum came up.

0:35 And I looked at this sum and I thought we've no chance of doing this sum.

0:38 But actually through a combination of uh high school maths,

0:44 we were able to do it.

0:45 So I thought might be fun to show you and and and on the way to show you

0:49 some of the tools of the trade that you

0:51 use when you're doing these maths calculations in theoretical physics.

0:55 So the sum is the following sum.

0:56 So I'll just call it I.

0:58 So I is so it's an infinite sum for the variable is N and it runs

1:03 from 1 to infinity and in the numerator it's N* S of NF and it's all divided

1:13 by a thing called alpha squared another sign except

1:17 this is sin squared now of 2 pi n/ alpha and that's then plus<unk> 2 n^ And I'm

1:28 interested in the limit of f going to zero.

1:32 That's what I need.

1:33 Alpha is just a constant which we don't need to worry about.

1:36 It's tells me some information about the black holes.

1:41 So this is the sum I've got.

1:43 And I looked at it and I thought I've got no chance of doing this.

1:47 So that just emerged from other work you were doing.

1:49 Right?

1:50 This is actually um we need this in order to help us

1:54 uh calculate a probability and probabilities you know always add up to one.

2:00 You can't have more than a probability of more than one.

2:02 And to guarantee your probabilities add up to one

2:06 you have to we call it normalize them.

2:08 And this is helps normalize the probability.

2:11 Now naively you might think this is zero because a sign of zero is zero.

2:19 And so when f goes to zero, this sign is going to zero.

2:22 But this is a bit more subtle because there

2:25 are very very large values of n in here.

2:28 And so you've got a very large number times a small number and it's it's

2:33 quite tricky to take that limit and this will turn out not to be zero.

2:37 So what we did was I didn't think we could do anything with it.

2:41 So I actually we just had a quick look at it on the computer.

2:44 So I'll just show you what we began to see.

2:47 So this is one over this factor I.

2:50 Okay, this is prefactor we're calling it.

2:52 And I did a sum the green line, the orange line,

2:55 the blue line and the red line are

2:57 for different values of this upper value of n.

3:00 So n of 10, n of 20, n of 50.

3:04 And you see what was happening as f decreases.

3:08 Remember I needed to go down towards zero.

3:11 Then say if I take the green line, this line was coming down and it seemed to be

3:14 going to this constant and then it shot off.

3:17 And of 20, same thing was happening coming down to this dashed

3:21 line and then it shut off and and it it

3:24 was shooting off earlier and earlier but as n increased

3:29 this was staying on this constant line and then eventually going off.

3:33 Remember n really has to go to infinity.

3:35 And it made me think, well, there's this looks like it's a number of order six.

3:39 So then I thought okay perhaps I can bound this it

3:43 might might give me an idea of what to do.

3:45 Bounding things is something we do at school.

3:47 We ask is something less than one value and is it bigger than another value.

3:52 So I thought I would just do that first.

3:54 So that's what I did.

3:55 And the way I did it was think about the denominator.

3:59 So that's this term here.

4:00 So if we think about this denominator.

4:04 So what is it?

4:05 It's alpha^ 2 sin^ 2<unk> n/ alpha +<unk> 2 n 2.

4:19 Now this here is always positive everything squared.

4:23 So that means that this thing is always going

4:28 to be um greater than just this term by itself, right?

4:34 Because or greater than or equal to it because

4:38 if this was zero then it would be the same.

4:41 But otherwise it's always greater because that's always positive.

4:45 But I want one over that.

4:47 So it means that one over one over this will always be less than one over that.

4:53 So it means that I can now say that I will always be less or equal

5:01 to the sum of n= 1 to infinity of n sin nf/ now I'll just include this term.

5:11 I don't I don't include that.

5:13 So that's p<unk>^ 2 n^2.

5:16 And you can see this looks what is this actually equal to?

5:20 Well, one of the n's cancelled, right?

5:23 So, this becomes 1 /<unk>^ 2 times the sum of n= 1 to infinity of sin nf over n.

5:37 Well, this looks a little bit better.

5:39 Still a nightmare.

5:40 We all have our tools of the trade, right?

5:42 You're holding your tools of the trade.

5:45 My dad was a joiner.

5:46 His tools of the trade were hammers and saws.

5:50 This is one of our tools of the trade.

5:52 It's the book of table of integrals, series and products.

5:56 An excellent Christmas present.

5:58 So in here are integrals and but there are some sums and one of the sums.

6:05 So you just got to get the right one is there.

6:08 Do you see what I want?

6:09 I want sin nf/ n.

6:12 Well, this is sin kx over k.

6:15 It's exactly the same thing.

6:17 So this turns out to be 1 /<unk>^ 2 *<unk>/ 2- f/ 2.

6:25 Okay, that's what that sum is.

6:26 But I'm interested in the limit of f goes to zero.

6:29 And now I can take this limit.

6:30 There's no ends multiplying it to cause me a problem.

6:34 So it means that in the limit of f goes to zero,

6:41 I must be less than or equal to and what I have is 1 over 2 pi.

6:46 That's the first.

6:47 So that's an upper bound on what this sum is.

6:51 And one over 2 pi is what?

6:53 6 point 3.14 6.28.

6:57 And that thing was towel.

7:00 Tao.

7:00 This is okay.

7:04 Now let me try and get a lower band.

7:07 And once again I go back to my denominator here.

7:10 To get my lower band I can make use of the same thing.

7:13 If I once again look at alpha sin^ 2 once again the denominator 2 p<unk> n/

7:20 alpha +<unk> 2 n^ 2 well sin^ squar is always less than or equal to 1.

7:31 Right?

7:31 The sin squar function looks something like this.

7:35 It starts at zero and it will go up to one and then it

7:38 will come down and it'll go up to one and it will come down.

7:41 So this is always less than or equal to one which

7:44 means that this thing will always be less than or equal

7:50 to alpha 2 +<unk> 2 n^ 2 because this thing is always

7:57 less than or equal to one and that's having equal to one.

8:01 So I can now do the same thing.

8:02 I can now say that I is now greater than or equal.

8:06 So this sum from n is 1 to infinity of n sin nf over and now it's this thing

8:15 alpha 2 +<unk> 2 n^ 2 and so I just want to write this if I pull this pi^2 out.

8:24 So this is equal to 1 /<unk> 2* the sum from n is 1

8:30 to infinity of n sin nf over n^ 2+ alpha 2 over p<unk>^ 2.

8:39 I've just pulled the pi squ out and so it

8:42 divides that alpha squ and leaves me with that n^ 2.

8:45 So I've got this sum.

8:47 So going, okay, it's looking better than the original,

8:49 but it's still But I go back to my favorite book, the Christmas present,

8:55 Christmas present book, and I see that on the next page, I have this formula,

9:01 the sum of k sin kx, sum of n sin nf over k 2+ a 2/ n 2 plus this thing.

9:12 This is equal to a a fixed function as well.

9:16 It's it's a known function.

9:17 So this actually is equal 1 /<unk>^ 2 into<unk>/ 2* by a shine

9:25 not a sign shine of alpha minus f alpha/ pi/ shine of alpha.

9:35 So that's what this sum is equal to.

9:38 Remember I is going to be bigger than or equal to this sum.

9:41 But remember I want the limit of f going to zero.

9:44 So I can once again set f to zero.

9:47 And can you see that I then get shine alpha divided by shine alpha which is one

9:54 because it's one thing shine shine of alpha

9:57 divided by shine of alpha is the same thing.

9:59 You divided shine of alpha by itself and so it just gives me one.

10:02 So this tells me that I must be greater than or equal

10:07 to remember I'm looking at the limit of f goes to zero.

10:12 The shines cancel and this becomes 1 over 2 pi.

10:15 So I've got that I must be both less than or equal to 1 over

10:19 2 pi and I must be greater than or equal to 1 over 2 pi.

10:25 There's only one way this can be if I equals 2 pi.

10:29 So I've actually done this sum without doing this sum.

10:33 Well done.

10:35 Which means that I equals 1 over 2 pi.

10:40 And so if you then go back to this prefactor which is 1/ i.

10:46 So 1 over i would be 2 pi which would be 6.1.

10:50 And that's pretty much what that is.

10:53 So for people watching you did this sum but this isn't

10:57 this isn't the number of primordial black holes in the universe.

11:00 It's not one over two pi primordial black holes.

11:03 This is just a little tool.

11:04 It's a tool on the way.

11:05 It was just a an important part of a calculation which will allow

11:08 us to then go and work out this probability distribution function in principle.

11:13 And how many primordial how was I know you don't know.

11:16 Haven't been able to solve it.

11:17 Haven't been able to solve the next part of it yet.

11:21 Yeah, that's ongoing.

11:23 That must have been quite satisfying when that happened.

11:25 Oh, it was fantastic because it never happens.

11:28 Usually when you put a bound on it'll tell you that okay I has got

11:32 to be less than some massive number and bigger

11:36 than a really small number and you know

11:37 it's a number of order for whatever one over 2 pi and it you you can't

11:42 do that much with it but to have it bound like that it was just chance.

11:47 That must also give you a feeling you're on the right track.

11:50 Yeah that's right.

11:51 And so it actually made me think, okay, I'm going to try and do this.

11:56 And and the thing that I I thought might help was if I

12:01 So you've got this sin squared in the bottom in the denominator here.

12:07 If I first of all brought it up into the numerator,

12:11 then you've got s time sin squares and and higher order terms.

12:15 And then there are things we learned at school high school about

12:19 trigonometric identities which I thought to myself if I can get it so

12:24 it's of the form like these which is sign over something and then

12:28 here you've got sign over something then I thought maybe may maybe I

12:33 can then make use of these identities but in fact something even nicer

12:37 happens which means I I didn't have to do any sums in the end

12:41 I have to write things out as sums but I didn't actually

12:44 have to do them and um we can do that if you want.

12:49 So you're going to what are you going to do now?

12:50 You're going to prove something.

12:51 I'm going to prove it.

12:52 What are you going to prove?

12:53 I'm going to prove this result.

12:54 I mean at some level I've proved it, right?

12:56 I've bounded it already and I've said it's got

12:59 to the upper bound and the lower bound are exactly the same.

13:02 So it's got to be this.

13:04 Is that not enough?

13:05 Why Why do you Why would you even prove it?

13:07 Like no, I was a bit nerdy.

13:08 I think I just wanted to try and convince myself that I

13:12 could do this uh because it was such a nice result.

13:15 I mean, two pies.

13:17 You you you always have a feeling you're on the right

13:19 track if you've got a two pie hanging in there.

13:21 And uh so that's yeah, that's why I then went and had another go.

13:26 That's it.

13:26 The second part of this video will be posted over at number file 2.

13:30 There are links below, but be warned.

13:33 This gets a little bit hairy, but but hang in there.

13:37 And while I'm here, can I take this opportunity

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14:00 Thanks a lot, everyone.

14:02 We'll be back soon with another video.

14:14 So I can do I can write this now as 1