 Hi, I'm Zor. Welcome to Unizord education. We continue talking about limits for sequences right now. We will switch to functions a little later, so it's limits for sequences and today I would like to talk about sequences which are infinitely small, so to speak. They're called infinitesimals. These are basically sequences which have a limit which converge to zero. Now this lecture is part of the course of advanced mathematics for teenagers and high school students. It's presented on Unizord.com. I suggest you to watch this lecture from this website because it has very nice notes, very detailed for every lecture and then registered students can take exam, for instance, and basically do the whole educational process. It's a course. It's not like an encyclopedia where you can just get to this particular topic or that. It's actually related and I do suggest you to take the whole course. So, back to infinitesimals. They are playing rather special role among all the different converging sequences. I mean, there are sequences which converge to one, to 25. Now, why did we decide to to separate sequences which converge to zero in a special category, so to speak, and we have this special name, infinitesimals. Well, actually there are very important role which these particular sequences, converging sequences, are playing, not only in mathematics, actually in many different subjects like physics or chemistry or anything like that. So it's very, very important. All right, so let me just go into the details. Whenever somebody is talking about, I would say something like infinitely small variable. Now, it doesn't really mean that they are talking about a particular number, however small it is, like one millionth. No, one millionth is not infinitely small. It's not infinitesimal. It's just a small number. Now, when we are talking about infinitesimals about infinitely small numbers, we are actually thinking about a sequence which takes values, which are getting closer and closer to zero. So it's a convergent, convergent sequence, and this particular convergent sequence has a limit of zero. So it's very important to understand that even if in a conversation somebody is talking about, okay, let's take this infinitesimal. It doesn't mean that they take a specific number. It means they are talking about sequence of numbers which are getting closer and closer to zero. They're talking about process, if you wish, during which certain value takes, again, sequence of values closer and closer to zero. So infinitesimals are not numbers. These are sequences of numbers. These are mathematical equivalent of a process. So something is supposed to change, like for instance, if it's a sequence, then the order number. As order number increasing, the values are decreasing to zero by absolute value. Okay, so that's done. Next. Yeah, there are a couple of examples, actually, of day-to-day conversation and it involves infinitely small numbers. And let me just give you this particular example. When we're talking about something like infinitely small distance between two objects, what does it mean? Well, it means that objects are involved in movement and the distance is getting smaller and smaller and smaller and as time goes on, this distance goes closer and closer to zero. So again, it's not a concrete distance between two objects. If we're talking about infinitely small, it means they are closing. It's a process. Okay, now after all, all these kind of introductory words, infinitesimals are just regular sequences which have the limit. It's convergent sequences, which means everything we have learned about convergent sequences actually is true for infinitesimals as well. And here are the major properties. Now, if there is a value of a sequence which goes to zero, sometimes if we are talking about infinitesimals, we might actually use one particular letter instead of having like curly brackets and index, etc, etc. So when I'm saying that epsilon is an infinitesimal, it means that it's actually an infinite sequence of numbers convergent to zero. That's what it means. So from now on, I might actually skip this part and just leave this saying, okay, let's consider that epsilon is an infinitesimal and that's how I will write it. Again, I assume that you understand that this is actually the sequence behind this one symbol. Then therefore, this is the sign for therefore, if k is some kind of a constant, then k times epsilon is also infinitesimal. Doesn't matter what kind of a constant this is, but if the limit of this is equal to zero, the limit of this is equal to, according to the properties of limits, k times the limit of epsilon, right? And since the limit of epsilon is zero, I will get zero as a result. So if epsilon is infinitesimal, k times epsilon, where k is a constant, is also infinitesimal. Next. If epsilon is infinitesimal and delta is another infinitesimal, then their sum also is infinitesimal. Again, it follows from properties of the limits. If limit of this is zero, limit of this is zero, limit of sum is equal to sum of limits and sum of zero plus zero is still zero. So the sum of infinitesimals is infinitesimal. Now, similarly, product, because of exactly the same logic, limit of the product is product of limit. In this particular case, when these limits are existent and they are convergent variables and zero times zero gives you zero. So these are kind of obvious properties. Now, what is left is this. And this is a whole different ball game. Because now we have something which has a value limit of zero and another which has limit of zero. I cannot say that one over delta, limit of one over delta is equal to one over limit, because that theorem was proven only for limit not equal to zero. In this case, limit is equal to zero. So we don't really know what it is. I mean, we cannot really refer to the theorem about limit of something is equal to something of limits, right? Like limit of sum is sum of limits. So this is not working this way. So what is this? Well, let's just consider a couple of examples and I will just show that it's not such a simple case. So first of all, let me just make a very simple example of these two infinitesimals, the ratio of which is a constant. Well, very easy. For instance, epsilon is, epsilon nth is equal to, let's say, 13 over n, epsilon, no, not epsilon, now it's delta. And delta n is equal to, whatever, 37 divided by n. Now, both of them are infinitesimals, right? Because obviously this is converging to zero as n goes to infinity and this, as n increasing, is also converges to zero. Now, what is the ratio? Well, ratio is equal to 1337, not changing, actually, as n is changing. So it's a constant. So the sequence which results in this is 1337, 1337, 1337. So obviously this is a convergent sequence which is constant and the limit exists and it's equal to 1337, right? So in this case, what we have is that epsilon over delta is just constant. Now, let's consider slightly more complicated case. For instance, this is now, what is epsilon over delta? This divided by this, well, this divided by this is 1337 and this divided by this is n squared here, 37 here and here. So we reduce by n here and here. So we have 1 divided by 37n. So it's no longer a constant. However, obviously this thing is convergent to 1337s because this one is converging to zero and the limit of sum is equal to sum of limit in this particular case. And this is again 1337s and this is zero. So first we have equal to constant. Secondly, we have that it's not equal to constant but convergent to a constant. Okay, let's go to another extreme. How about this? Now, what do we have here? We have equal 13n squared divided by 37n. So it's this which is obviously converges to zero as n is increasing which means that in this case epsilon over delta is also in infinite, infinitesimal. Right? Also convergent. Next example. How about this? So the epsilon n over delta n is equal to 13n 37n squared. We can reduce by n and what happens? What is this thing? Well, it's increasing. It's monotonously increasing and it's basically limitlessly increasing to infinity. So as n increasing and 1337 times n also increasing and basically it means that it's not converging to any number. It's not a convergent sequence. Although we can say that it actually tends to infinity but we will talk about infinity separately. There is no such number as infinity more or less. So this is not a convergent sequence at all. So we had convergent, non-convergent and however, we can say something about infinity in this particular case. And let me give you an example when it's just really like non-convergent at all. It doesn't concentrate to any particular number and doesn't go to infinity. Let's say this is this but I will also multiply it by minus 1 to the power of n. So what happens? Epsilon over delta is equal to 1337 and here and there it will be reduced times minus 1 to the nth degree. Now, what does it mean? Well, it means that all odd n would give me minus 1337 and all even n will give me plus 1337, right? Because minus 1 to the even degree is plus 1. Minus 1 in the odd degree is minus 1. So this thing is a sequence which goes minus 1337 and plus 1337. This is 0. So this is my sequence. It goes here, here, here, here, here and obviously this is not converging to anything, right? So this ratio of two infinitesimal, infinitesimal, can give you anything. It can give you a convergent to something. It can even give you a converging to 0, which means it can be an infinitesimal. It can be non-convergent, but monotonously increasing, limitlessly increasing. So we sometimes can say about infinity as being the limit it tends to. And finally, we can have a situation when it's definitely non-convergent at all. It doesn't really go anywhere and it's within certain boundaries, but it jumps back and forth, back and forth, which means there is no limit. So I was just trying to explain that this is indeterminate by itself. Without additional transformations, it's indeterminate. And I did have some transformations. I mean, whenever I had, for instance, n squared here and here, I was actually reducing by n and then as a result, I had some formula, which basically can be analyzed as having or not having a particular limit, convergent and non-convergent sequence. So this by itself is an indeterminate value. It needs some transformation to make a final judgment about convergence of this. Now, speaking about practical situations, you see, again, I made this specific lecture about infinitesimals because they are really very, very widely used. In mathematics, whenever you are doing some kind of analysis of functions, in physics, for instance, the concept of speed. What is a concept of speed? Well, you are at certain point. The moving object is moving passing you. Now, how do you know the speed of this particular moving object at this particular moment in time? Well, you stop the watch at moment. It's actually passing you. Then you start the watch. Then you stop the watch after a certain period of time. Now, this object covers certain distance, right? And then you basically divide the distance, which it covered by the time which took this particular object to cover this distance. And you have an average speed during this period of time. If you would like to have a more precise speed at the moment when it's passing you, well, you have to reduce the time. So the closer your time becomes to the moment when it's really passing you and the smaller the distance actually will be, the more precisely the result of this division will correspond to a speed at the moment it's passing you. So what are we talking about? Obviously, we are talking about ratio of two infinitesimals. One infinitesimal is the distance and another is the time which took this particular object to cover the distance. So the smaller the time and the smaller the distance, the more precise their ratio would correspond to the speed. So this is exact example of ratio between two infinitesimals. And in a good case, like real movement, we will have some kind of convergence of this ratio to a certain value which actually is a value of the speed at that particular moment in time. It's a limit, so to speak. So there is no such thing as a speed without the concept of a limit because the object is moving with different speeds, obviously. So you can obviously talk about average speed, but if you want to talk about speed at any particular moment, you need to resort to infinitesimals. So from that moment, you step forward by certain value, measure the distance, measure the time, have the ratio, and then you try to reduce this distance, which means you have to repeat the experiment, I guess, under the same conditions. But that's not mathematics, that's physics. Alright, so I just wanted to explain how important infinitesimals are. If even such a simple concept as a speed at a particular moment in time really requires usage of this concept, and in this particular case, the ratio. So the ratio between two infinitesimals is probably the most important part of these manipulations with infinitesimals because all other operations like addition and multiplication is trivial, basically. This is not trivial. Alright, so that's it for today. I recommend you to read the details, the notes for this particular lecture on Unisor.com. And basically, that's it. Thank you very much and good luck.