 Hi, I'm Zor. Welcome to Unizor Education. We continue talking about limits of sequences, and today's topic is infinity. Now, as usually, I do recommend you to watch this lecture from the Unizor.com website, because it contains detailed description and notes for every lecture. Plus, registered students can take exams, for instance, which is very, very interesting. The site is free, and you can take as many exams as you want, basically. Alright, so, infinity. I would like to start with something which basically should be inculcated in your minds. In mathematics we are talking about, and this is the mathematics on the level of high school, basically. There is no such thing as infinity. I want to start with this particular slogan, there is no infinity. So, what is infinity and we are talking about? Well, casually we are talking about infinity, and we mean something. But as everything else in mathematics, infinity is abstraction. So, in this particular case, well, number is also an abstraction. Function is also an abstraction. But infinity is an abstraction of even higher degree, because with function we can have a graph, a formula. I mean we have something more than just properties. With infinity we have only properties, and it's not even the properties of infinity, it's the property of sequences which are tending towards infinity. Not converging, again, I would like to warn you against using the word convergent as applicable to sequences which do converge to certain real limits, like real number. Those which go to infinity, we should not really speak about converging to infinity. So, what exactly is infinity and what do we mean? First, let me just give you an example of a sequence which basically can be very well qualified as going to infinity, so to speak. And here it is, for instance, like 2 to the power of n sequence. Obviously it's increasing, and gradually it becomes greater and greater than anything we would like to fix as its boundary. So, it's boundless, it's limitless. And what also is important is that after it crosses certain boundary, it stays above that boundary, which is actually very similar to the concept of a regular limit when we are saying that the sequence which converges to a limit gets however close to that limit after a certain order number and then stays within that particular distance from that limit. No matter how small a distance we can set before this process. So, for any however small distance from that limit, we will always be at certain moment in our sequence's life closer to that particular border and stay close. So, in this particular case, it's above any border or boundary and it will stay above that boundary all the time after we cross it. So, we are actually approaching the definition of a sequence which limitlessly grows. So, let's just try to formulate it. Okay, the formulation is something like this. For any number A, real number A, concrete number A, there is an order number N. This is the order number after which we are saying we are crossing this limit, such that XN would be greater or equal than N if N greater or equal than capital N. So, all sequence numbers with numbers greater than this N will be above our border line A. Now, it doesn't mean like in this case that our sequence is monotonic. No, it can grow and it can go down again, but then it will go even further and move a little bit down. It's also quite possible. So, not only this type of movement, but also this type of movement. However, in both cases, for any limit A, we can always find this point, this number N, after which we will be greater forever above that level. We will be above that level. That's actually what we mean when we are saying that our sequence is growing infinitely. I used the word infinite in this case. And basically, it's nothing more this statement about this particular sequence XN, which grows infinitely or infinitely grows. So, this statement is nothing more than this. So, whenever I'm saying that certain sequence is infinitely growing sequence, it means that for any number A, I will always be able to find order number capital N such that all members of that sequence will be greater or equal than A the order number is greater than that capital N. But this is a very long statement. So, instead of saying this, we are saying our sequence grows infinitely. In exactly the same way as we are saying that our sequence has a limit A, which basically means that for any however small distance from the A, there will be found a number and after which our sequence will stay closer than that particular distance to a limit A. So, it's exactly the same thing. So, using the word infinity or infinite in this case is nothing more than a shorthand for a bigger and more clumsy, if you wish, definition of what exactly this means. So, consider the word infinity as a shorthand for describing the process which is infinitely growing. Now, we obviously have a symmetrical concept of negative into infinity. What does it mean that our sequence is infinitely decreasing? Well, or decreasing to minus negative infinity to negative infinity. What does it mean? Well, it means basically very similar to this except I have to put this sign in reverse. It would be lesser or equal to any number A and A can be any, however, any, any. That's what's important, this any. It can be any, which means however small in a negative sense, I mean however big in absolute value was a minus sign, right? So, no matter how far down to the left of the x-axis I will go, as a border I will always find such a number after which my sequence members will be to the left of this particular number A and stay to the left forever for all other and greater than capital N, all right? So, these are definitions of infinity which sometimes is called positive infinity and negative infinity. Now, is it applicable as far as syntax is concerned to write something like this? Of 2 to the power of N as N goes to infinity equals to plus infinity. This is the sign for infinity. Well, quite frankly, I don't like it. This equal sign, which means it just implies that this is a number. So, probably people will understand you what you mean. I would rather prefer something like this. It's increasing to infinity or plus infinity. Sometimes you can say this because this implies monotonic increasing. This implies just moving towards infinity. But in any case, it doesn't really matter how you describe it syntactically. What's the meaning of this is? It is that for any, I will use mathematical character symbol for the word for all or for any. For any number A exists, this is the mathematical symbol for exist. Exist number N such that if order number N is greater than capital N, immediately follows that XN greater equal to A. So, that's what it means. And then, yes, we can put it this way. So, this means exactly the same as this. It's just shorter. And in conversation, you can obviously say that something goes to infinity or grows to infinity or decreases to minus infinity or increases to plus infinity. Well, so basically, that's my explanation of what infinity is. And let me repeat, there is no such thing as infinity. It's not a number. It's basically an assumption that there is a certain process which is growing in such a way that no matter what kind of a border we will establish, it will be above that border. Or if it's decreasing, it will be below that border. So, it's limitless growing or limitless decreasing. That's what it basically means. And now, let me give you just a few examples of sequences. Okay, sequence number one. N squared plus one divided by N. Okay, what's important is that this is a polynomial of the second degree and this is the polynomial of the first degree and the greater the degree, the faster it grows. In this particular case, this goes faster than this one, which means we should really expect it to go to infinity. Well, obviously, you can divide it and it would be the same as N squared divided by N, which is N plus one Nth. As a side note, this is a monotonic sequence, right? With N is equal to one. Plus one over one, which is two. With N equals to two, it's two plus one half, which is five halves, right? With N equals to three, it's three and one-third, which is what? Ten-thirds. And these are increasing. All right? These are increasing. So, it's monotonically increasing and it's kind of obvious because as N increasing by one, this thing is increasing by one and this one is decreasing by some fraction, right? From one-second to one-third or from one-third to one-fourth. So, this is decreasing, this is increasing, but this increases always by one and this is by a fraction of one. So, which means we're always increasing as N grows. Now, how can I prove that this thing is infinitely growing? Well, let me just do it by definition. So, for any A, however big I choose the number A, I should find the number N such that if my lowercase N greater or equal than capital N, my sequence would be greater or equal than A, right? This is XN. Okay, so let's fix number A. However large, doesn't really matter. And I would like to find number N. So, I would like to find number... Basically, I would like to resolve this inequality. N would be a solution to this inequality, right? Can I solve it? Does it have a solution? Because if it has a solution and using the fact that this is monotonic, it means that if I will find one particular N where it's true, then everything higher than that would be also as well true, right? So, now N is positive so I can multiply both by N. I will have N squared plus one greater or equal than A N. Or I can subtract positive number, well, actually any number. It should be greater than or equal to zero. Now, what is this? This is a quadratic polynomial. Quadratic polynomial. Now, the coefficient with N squared is one. So, on a graph, it would look something like this, right? This is the parabola, which means that it's equal to zero at these points. And it's greater than zero in this area, right? And this as well, but we are not really interested because we are interested in increasing numbers N, right? So, the solution to this is what? A plus minus square root of A square minus four divided by two, right? These are two solutions to the equality with equal. So, which means that if I will take the larger solution, which is this one with a plus. And so, if N is equal to this, then it would be exactly equal. But N is supposed to be an integer number, so I shouldn't really say this one. This might not be actually an integer. But the next integer after that, right? So, if my capital N is chosen as greater than this one, then my sequence with numbers lowercase N, which is equal to capital N and further, will be greater than zero, right? So, that's how we solve it. We have proven that for any A, we have found this particular value such that if my index, my order number is greater than this number, my sequence members will be greater than A forever. It will stay forever because it's monotonic actually. Okay, that's just one of the examples. And we have proven that the limit of this thing is infinity, or which is probably a better formulation, it's gross to infinity. Limitlessly gross to infinity. All right, another example for negative infinity. Okay, negative infinity. I have tangent of minus p and minus pi N to N plus one. Now, why did they choose this way? Well, let me just think about it. I wanted basically something which grossed to negative infinity, right? So, I remember that the tangent has a graph like that. So, if I will reverse it negative, I will have this. So, I would like to, as N is increasing, I would like actually to move this way. Then my tangent would be going to negative infinity, right? So, what are these points? Well, these are the points. Well, minus to make it this way instead of original this way. Now, pi over two, this is pi over two. But I multiply it by N divided by N plus one, which is slightly less. The bigger N is, the closer N over N plus one goes to one, which means my point is getting closer and closer to pi over two, but never equal to pi over two because at pi over two it's not defined, right? So, basically, that's the origin of this particular sequence. Now, all we have to do, we have to prove that we have, for any level A, negative in this particular case, we can always find the point after which all these tangents will be less than A, right? Well, again, this is definitely a monotonic sequence because tangent is monotonic function. We know about that. So, if we will find the point where this tangent is equal to A, then all larger numbers will give us basically closer and closer, they will be less than A and closer and closer to pi over two because the tangent being decreasing to negative infinity, right? Okay, so let's just find, for any fixed number A, negative in this particular case, we will find the point where this tangent is equal to A. And again, it will not be an integer number and is an integer. So, basically, after that, integer number would be our number, which we're looking for, the number after which our sequence will be less than A. So, let's just solve this. Tangent of minus pi n over n plus 1, 2 times n plus 1. Let's solve when it's equal to A. Well, first of all, obviously, tangent is odd function, so minus can be here, right? Now, instead of that, I will put minus here. So, that's an equivalent equation. From now, we can find that pi n over 2n plus 1 equals to arc tangent of minus A. Sometimes people use tangent in minus first power, but arc tangents is something which, basically, I'm more used to. A function inverse to a tangent, all right? So, there is such a number, obviously. And now, all I have to do is to solve it for n, right? So, what's the solution? Pi n equals to 2n arc tangent of minus A plus 2 arc tangent of minus A. So, n goes here. Okay, so what do I have? n equals to 2 arc tangent of minus A divided by pi minus 2 arc tangent of minus A. Now, let's think about it. As A grows negatively, which means it's decreasing, basically, more and more. Minus A would be increasing more and more. Now, what is arc tangent? Now, arc tangent is the angle tangent of which is equal to this, right? So, if this is growing to infinity, my arc tangent grows to pi over 2 times 2 is pi. So, these two things are becoming closer and closer together. Now, this is constant. So, as we go closer to A, my denominator is getting smaller and smaller, closer to zero, right? And this one is relatively close to pi. So, as A increasing to a negative, well, increasing by absolute value. So, minus A is increasing towards plus-plus infinity. As it happens, this thing is closer to pi, and this one is getting closer to zero. So, that's why this particular number is getting greater and greater. But anyway, it's a concrete number. So, we can choose N to be greater than this particular expression. Choose integer N, obviously, because this is most likely is not integer. So, we choose this integer N, and for all lowercase N greater than this one, I will have my tangent to be below level A, which is negative level A, no matter what A is. Again, we have proven that this particular sequence is decreasing to minus infinity, because for any level A, we can find the number after which we will be below that level. And final example is something which is not a sequence which goes to plus or minus infinity. However, it grows by absolute value. To make up this example, I started with something which does not have a proper limit, which is sin of N. Now, sin of N is going up and down, up and down, up and down, as N is increasing. But I would like actually to get infinity involved, so I will multiply it by N. So, sin is basically bouncing between plus and minus one, but this one is increasing, infinitely increasing, obviously. So, if I will consider this particular sequence, it will behave something like this, higher and higher up and higher and higher down. So, it will go to, well, plus infinity on this particular pieces. Then to minus infinity on these particular pieces, which means we don't really go to any kind of a limit. We cannot say that this is infinitely growing or infinitely decreasing sequence. It's just sequence which does not have any limit and it doesn't grow or decrease. It basically does both, which means it doesn't really qualify for being called increasing or decreasing. If we choose, for instance, any number A and we would like to find such a number in this sequence, after which it's greater than A, we will not be able to find it. Because after any number, we can find, obviously, the wave which goes above it, but then it will go below. Our condition was not only it should cross that border, but it should stay over that border or below that border if it goes to minus infinity, right, to negative infinity. So, again, same thing as there are cases with limits. If you are approaching a limit and then go out from it, again approaching and then go out, it's not a convergent sequence. It means just the sequence which does not have any limit and it's not convergent. This also represents an example of the sequence which has absolute value of its members is growing infinitely. Absolute value, but the sequence itself is going up and down, up and down, which means it doesn't really infinitely grow nor infinitely decrease. Okay. That was it about infinity. And remember, there is no such thing as infinity. It's just a short form for expressing our characteristic of certain sequences that they are directionally going into some direction, positive or negative, and they are limitless. So, again, the bottom line is we are talking about direction and unlimited sequences, directions up or down and limitless. And that's why we are actually getting into this word, basically, infinity, which means nothing more than whatever property of the sequence I have just explained. You cannot add infinity or divide infinity, although you can symbolically say that, well, maybe there is some kind of an operation between two sequences which are, let's say, for instance, you are subtracting one infinitely growing from another infinitely growing. Well, there are many cases and there are different results which might result in this. This is something which we will call indeterminate sequences. And it needs an effort to find out whether this sequence, which is basically a difference between two unlimited sequences, maybe it has some limit, maybe it doesn't. Same thing if you divide one by another. I mean, if you have two infinitely growing sequences and you divide one by another, well, you can have different results. It all depends and it all requires special attention, which we will definitely spend some time for. And that would be a subject of the next lecture where we will talk about these indeterminates. That's it for today. Thank you very much and good luck.