If you were to define what it meant to be a banana, you had to tick every box on that list and then you could be nothing but a banana. We’re going to look at a sequence and show that it converges to a limit you weren’t expecting. So the sequence is as follows: the first term is 9, second term is 99, third term 999. Fourth term - you guessed it - four 9s; and we’re going to continue with this pattern. So obviously this sequence goes to infinity. We’re adding more and more 9s, so in the usual sense this is a divergent sequence; it goes to infinity, it’s getting bigger.
But I’m going to show you that in fact this tends to -1. This sequence of adding more and more 9s, making the number larger and larger, in a certain situation can be shown to tend to -1. And here’s how it works. First of all we need to establish what we mean by saying a sequence tends to a particular number. This might seem like something you know, or you’ve seen before, but this is going to be really important in justifying why this is -1 so bear with me.
So if we start with a different sequence; so suppose I have 3/2, 4/3, 5/4, 6/5 and it’s going to continue. So we’re adding 1 to each of the numerators - numbers on top - and we’re adding 1 to each number on the bottom. So the nth term, if we would sort of go along and say pick a certain point in the sequence, what does the nth term look like at that position? It’s going to be n plus 2 over n plus 1. So first term, plug it in, we get 3 on the top over 2. Third term here this would be n equals 3, we get 5 on top 4 on the bottom. Whatever position down you might say to me, find the 1000th term in the sequence, it’s a 1002 over 1001.
Now this hopefully you can see is going to tend to 1. As we increase n, we go further and further along the sequence, the top and the bottom get closer and closer together. So we can get as close to 1 as we want by going further and further down the sequence. So the way we would show that is you take your nth term - which here is n plus 2 over n plus 1 - and we subtract off what we think the limit is, so here we think it’s going to be 1, and then we work out the distance between these two. So I’m calling this our distance function.
Now if we’re working in the real numbers - which these are all real numbers - then the distance function is just the modulus of one number subtract the other. So we take the modulus because it’s a distance so it should be positive - you can’t really have a negative distance. So here the distance in the real numbers is n plus 2 over n plus 1 minus 1. What is n plus 2 over n plus 1? Well we can simplify that and say, well that’s actually the same as n plus 1 plus another 1 all over n plus 1 minus 1. But then we’ve got an n plus 1 over an n plus 1 so that’s a 1 so we can say that’s actually 1 plus 1 over n plus 1 minus 1 so we get 1 over n plus 1.
So again, with this distance function the usual way of measuring distance, we can show that the nth term of the sequence and what we think the limit is; we can say the distance between them looks like 1 over n plus 1. So as we now go further and further down the sequence what we’re doing is we’re letting n tend to infinity. We’re making n bigger and bigger, what happens to our distance as n gets bigger, the distance tends to zero. So as n tends to infinity, the distance between the two things is 1 over n plus 1 which very clearly tends to zero, so we would say this sequence converges to 1. And we’ve done that using the distance function, which for the real numbers is just modulus of one of them minus the other. I’m going to need more paper. As always. The reason I’ve introduced the distance function, as some of you may have guessed, is if we’re going to show our sequence of 9s is tending to -1 we obviously can’t use the normal definition of distance because the distance between a larger number and -1 is getting bigger so it’s just not going to work, it’s not going to tend to zero. So we need to think about what other definition of distance can we use to show that this sequence is indeed going to -1. And the one we use is called the 2-adic metric - and we will check this is a distance. You say the distance between two numbers x and y. So x and y are going to be integers because here in our sequence we’re only working with integers so we’ll keep things simple.
So the distance between two integers in this metric is defined to be 1/2^m, where m is the largest power of 2 that divides x - y. So this feels a little complicated, so let’s look at an example, let’s think about what does this mean. Let’s suppose x is 100 and y is 4. So in the usual measure of distance we would say the modulus of x - y is the modulus of 100 minus 4, they’re 96 apart. Feels hopefully kind of obvious. Now in the 2-adic metric we’ve changed our definition of distance, so we need to think about what power of 2 divides their difference. So we already know the difference, the difference between them is 96. So if we think of powers of 2 we’ve got 2, 4, 8, 16, 32, 64. 128 is obviously too big so I’m going to stop there. 64, well that’s 2 to the 6, but 64 doesn’t divide 96. It’s smaller than it but 64 times 2 - which would be the next smallest thing you could have - wouldn’t give you 96. So we actually need to go down to 2 to the 5, which is 32, which perfectly divides 96. 2 to the 5 divides the difference between our numbers so, in our definition, we can say in the 2-adic metric the distance between 100 and 4 is 1 over 2 to the biggest power, so here 2 to the 5, so that’s going to be 1 over 32.
So usual definition of distance that we’re all used to, these numbers are 96 apart, 100 and 4. However in this metric they are 1 over 32 apart; which doesn’t feel at all natural and - (Brady: Or useful) - Or useful. I mean well that’s a whole other conversation. We’re entering the realm of pure maths. This - now that you’ve set me off on this - it is useful because we learn properties about the numbers. So this kind of Investigation, okay this isn’t a distance in our world, but we can learn things about distributions of numbers through this. Ao this 2-adic metric can be generalised and is used to actually in fields medal level research to learn about rational numbers. So beyond beyond my pay grade, beyond my ability to do the maths, but there is some usefulness in there.
So we’ve got our 2-adic metric. So I’ve just told you this is a way of figuring out how far apart two numbers are, but you need to check from a mathematical perspective that it does everything you would expect a distance measure to do. So we have a set of axioms. In order to be a distance function: it maps from two numbers in our starting space into the real numbers, we get an answer that’s a real number. Where d satisfies three things - this one’s called positivity. And these should hopefully fit with your intuition of what we might expect from a distance. If we have two numbers x and y, the distance between them is greater than or equal to zero. We’re talking about distances, okay they might not fit our usual way we think about them, but they still need to be positive. You can’t have a non-positive distance, negative distance makes no sense. And in fact you can say and it equals zero if and only if x is equal to y. So you only have a zero distance if it’s the same point. Again, hopefully fits with our usual notion. So if we subtract y we just get a(2 to the m).So if we
subtract this one, we just get minus b(2 to the m).So if we add this one and this one, we get a minus b
times 2 to the m.So what’s the biggest power of 2 that divides this? Well this is just 2 to the m.So we can say that our distance from x to z
has to be less than or equal to 1 over 2 to the m.So that satisfies the triangle law.So we’ve now checked that our 2-adic metric satisfies all three of the axioms, so we can be sure that it is indeed a valid distance. So
the distance between the nth term and the limit tends to zero as n goes to infinity.
We need to show that the distance between the $n^{th}$ term of the sequence and the limit, which is -1, tends to zero as $n$ goes to infinity. We demonstrated this with the example of $n+2$ over $n+1$, which tends to 1 as $n$ gets bigger and bigger. Therefore, the distance between the $n^{th}$ term and the limit tends to zero as $n$ goes to infinity. And they’re looking to hire those sorts of people. So if you’re interested in a career in finance, check out the Jane Street Careers page and see what they have to offer.
We can say that the nth term of the sequence is 10 to the n minus 1. The distance between the nth term of the sequence and the limit -1 is measured using the 2-adic metric, which is 1 divided by 2 to the n. As n tends to infinity, the distance between the two numbers gets smaller and smaller, converging to -1. This happens for any sequence, not just the one with 9s.
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