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The limit of a numeric sequence.

The lecture is devoted to the most important concept of mathematical analysis – the limit of a numerical sequence. And before we introduce a definition of this concept, let us look at what a neighborhood of a point is.

So, we are given a point, let us denote it a on the number line, and some positive number ℇ (Epsilon), in order to build a neighborhood of point a, we put intervals of length ℇ to the right of point a, and to the left of point a. We get an interval with ends (a - ℇ, a + ℇ). This interval is called the neighborhood of the point a. For example, the neighborhood of point 2 of radius 1 is the interval from 1 to 3.

What do we need to further study the definition of the sequence limit?

We should be able to turn from neighborhood to inequality. What does it mean that point x belongs to the neighborhood of point a? This means that x satisfies the inequality with modulus, and x minus a is less than ℇ in absolute value.

In addition, we will consider the concepts of neighborhood of infinitely distant points. So, ℇ still is a positive number, plus infinity is an abstract point and, in fact, it does not exist, it is located to the right of any real number, any point on the number line. This is how we have conditionally marked it (see Fig.).

So, what do we call ℇ neighborhood of a point plus infinity? Where are we going to lay it off?

We always have a point of origin on the number line and this is point 0. We lay ℇ off the point 0, in this case ℇ by the neighborhood of the point, we call the interval from ℇ to plus infinity. Similarly, we define a point minus infinity, which is a point lying to the left of any real number. To construct ℇ neighborhood, we put an interval of length ℇ to the left of the point 0 and get ℇ neighborhood of the point minus infinity, this is the interval from minus infinity to minus ℇ.

Generalizing these two concepts, we get ℇ neighborhood of an infinitely distant point of unsigned infinity, it is called the union of two intervals.

Again, we need the ability to turn from the neighborhood membership conditions to inequality. To belong to ℇ neighborhood of the point plus infinity, minus infinity, and unsigned infinity, means to satisfy the corresponding inequality (see the video).

Finally, we can now define the limit of a numerical sequence. The abstract logical form of this definition looks like this (see the video).

Your task is to learn to read this definition.

 Firstly, you should remember how it is written, and, secondly, be able to read it.

The definition is as follows: a number a is called the limit of the numerical sequence а_n if for any ℇ neighborhood of point a there exists such a natural number N that whatever the natural number n > N is, the condition а_n belongs to ℇ neighborhood of point a is satisfied.

The transition from writing the definition in the language of a neighborhood to the definition in the language of inequalities is made using those transitions that we have already mentioned, which means to belong to ℇ neighborhood of point a.

We can see how the definition is written in logical form (see on the screen). How this definition is read: the number a is called the limit of the numerical sequence а_n if for any positive number ℇ there is a natural number N that for all natural numbers n>N, the inequality modulus а_n minus a is less than ℇ is satisfied.

So, your task is to learn this definition, learn how to move from the definition in the language of neighborhoods to the definition in the language of inequalities, and read both of them.

Now let us speak about the sequence limit. а_n is equal to 5 for any n. This sequence is called stationary, all elements are 5,5,5, and so on. Let us prove that the limit of this sequence is 5. So, on the number line, we mark the point 5. It is highlighted in red. All the elements are here, there are infinitely many of them, but they are represented by a single point 5. In which case do we say that the number 5 is the limit of the sequence?

Let us turn to the definition. This means that whatever the neighborhood of point 5 is, all the members of the sequence must fall into it, starting from a certain number. Let us take ℇ neighborhood of point 5, this is the interval centered at this point. What can we see? Whatever the neighborhood of point 5 is, all the members of the sequence, and these are the 5s, fall into this neighborhood, but this means that the limit of this sequence is 5.

Let us take one more example. The sequence а_n is given by the formula of a common term – one divided by n, all the members of this sequence are different 1, 1/2, 1/3, 1/4, and so on, but what we notice is that as the number increases, the elements of this sequence become closer and closer to point 0. Note that there is no point 0 among the members of the sequence, but this number is the limit of this sequence.

Let us advance arguments. We have to prove that no matter what the neighborhood of point 0 is, starting from a certain number, all the members of the sequence fall into it.

This is not a strict proof, but an illustrative definition of the limit. So, we take a neighborhood of radius 1/2 of point 0, we see that starting from the third term (highlighted in red on the screen) 1/3, ¼, and so on, are in the neighborhood of point 0, if we take a neighborhood of a smaller radius, the illustration shows us again that starting from a certain number, all the members of the sequence will be in this neighborhood.

Not every sequence has a limit. Let the sequence b_n be given by the formula (see on the screen). Substituting natural numbers instead of n, we get the sequence -1, 1, -1, 1, and so on. On the number line, we mark two points only

-1 and 1. It is -1with odd indexes, it is 1with even indexes.

So, why doesn’t this sequence have a limit? Let us advance arguments why 1 is not the limit, in fact, at point 1 there are infinitely many members of the sequence. So, let’s take the neighborhood of point 1 of radius 1. This will be the interval from 0 to 2.

What do we see? If 1 were the limit, all the members would have to be there, starting from a certain number, but we see that no matter what number we take, there are infinitely many members with large indexes that are located at -1. There are infinitely many members of the sequence outside this neighborhood, thus, 1 is not the limit. Similarly, we easily argue that -1 and any other real number is not the limit of this sequence. As for this sequence, we say that its limit does not exist.

So, a numerical sequence can have a finite limit. In this case, we say the sequence converges, i.e., the limit is equal to the number.

We call all other sequences that do not converge divergent. So, we divide all the sequences into convergent ones, those that have a finite limit, and divergent ones.

The question arises: what can be the limit if it exists? We have seen that the limit can be equal to a number, and it is still possible that the limit is equal to infinity. So, the definition is as follows. A sequence is called infinitely large if its limit is equal to infinity. It can be with a plus, with a minus, it can be an unsigned infinity.

In order to write a definition in any of these cases, we refer to the general definition of the limit.

In the language of neighborhoods, each of these definitions is written and read in the same way as we read this definition in case when a is a number, so your task is to learn how to write down the definition and read everything correctly, based on what we have already discussed with you. Thus, we have just analyzed the concept of an infinitely large sequence.

There is also an infinitesimal, this is a convergent sequence, the limit of which is equal to a number, but a special number equal to 0.

So, we will briefly call a convergent sequence, the limit of which is zero, infinitesimal.

There is a connection between infinitesimal and infinitely large sequences, which is formulated in the form of the following theorem. If the sequence а_n is infinitely large, the sequence 1/а_n is infinitesimal, and vice versa, if the sequence а_n is infinitesimal, if there are no zeros among its members, 1/а_n is an infinitely large sequence. We will not consider the proof of this theorem.

Let’s turn to the properties of convergent sequences which have a finite limit.

The first theorem on the uniqueness of the limit of a convergent sequence. If the sequence converges, its limit is unique.

Let’s find out the structure of this theorem. The sequence converges, so by the condition, there is a number a, that is the limit of the sequence а_n. We need to prove that this number a is unique.

We will try to prove it. Suppose, on the contrary, that there is still a number b other than a, for definiteness let it be a<b. So, the number b is also the limit of this sequence by assumption. We form a positive number ℇ equal to b minus a in half, this is half the length of the segment from a to b, and build a neighborhood of points a and b of the specified radius.

Now we turn to the limit of a sequence а_n is equal to a, it means that starting from a certain number, all the members of а_n are ℇ neighbourhood of the point a, let’s say that all members with indices larger than some number N1. By assumption, the limit of the sequence а_n is equal to b. By definition, this means that all the members of the sequence are in ℇ neighborhood of the point b, starting with those numbers that are greater than N2.

And now we form the number N equal to N1 + N2. We understand that this number is the sum of two natural numbers which are greater than both N1 and N2, it means that a with this index is in ℇ neighborhood of point a since N is greater than N1, and is in the neighborhood of point b since N is greater than N2. There is a contradiction.

We have already built neighborhoods that don not intersect.

The conclusion is the assumption was made incorrectly, there is no other number equal to the limit of the sequence а_n, i.e. the limit is unique.

The second theorem is on necessary convergence conditions. If the sequence converges, it is limited. Let’s leave the evidence without arguments. In fact, this all follows from the definitions of the sequence.

The only thing you should always keep in mind is that the inverse theorem is not true, i.e., if the sequence is limited, it does not mean that it converges. We have already considered this example. The sequence -1 to the power of n, its members are -1, 1, and so on. The sequence is limited, it is closed by a segment from -1 to 1. We analyzed this example and said that the sequence has no limit. It is divergent.

The following theorem allows us to calculate the limits of the sequence in some cases. On the product of an infinitesimal and limited sequence, the theorem states that the product of such sequences is infinitesimal, i.e., the limit is zero.

Consider a simple example of the limit of the fraction sin(n) divided by n. We prove that the limit is 0. Let’s represent this fraction as a product of 1/n and multiplied by sin(n). The sequence 1/n is infinitesimal, we know its limit is zero. And sin(n) for any n is located between -1 and 1, so this sequence is limited.

Products of an infinitesimal by a limited sequence infinitesimal, the limit is 0.

Another property is related to the concept of subsequence. First, we define what it is. If we have a numeric sequence, what do we call its subsequence? We choose any strictly increasing sequence from the natural numbers, e. g, 1357, and so on. With these indexes, we select the members of а_n sequence. This will be the subsequence of this sequence.

Theorem 4 states that if a given sequence а_n converges, any of its subsequences also converges and has the same limit.

And the most important theorem 5, which we use in the calculation of limits, is as follows: if the sequence of а_n and b_n converge, i.e., have finite limits, the sum, the difference, the product and fractions of these sequences with some restrictions are converging, and the limit of the sum is equal to the sum of the limits a + b, the limit of the difference equals to the difference of numbers a and b, the product of a and b, the only point about the fraction is different, there is a limit to the number b, it should not be 0, i. e., the sequence b_n is not an infinitesimal sequence