What is the name of the sequence? How to calculate the limits of sequences? Examples of sequences converging to a finite number

Mathematics is the science that builds the world. Both scientists and ordinary people - no one can do without it. First, young children are taught to count, then add, subtract, multiply and divide; by middle school, letter symbols come into play, and in high school they can no longer be avoided.

But today we will talk about what all known mathematics is based on. About a community of numbers called “sequence limits”.

What are sequences and where is their limit?

The meaning of the word “sequence” is not difficult to interpret. This is an arrangement of things where someone or something is located in a certain order or queue. For example, the queue for tickets to the zoo is a sequence. And there can only be one! If, for example, you look at the queue at the store, this is one sequence. And if one person from this queue suddenly leaves, then this is a different queue, a different order.

The word “limit” is also easily interpreted - it is the end of something. However, in mathematics, the limits of sequences are those values ​​on the number line to which a sequence of numbers tends. Why does it strive and not end? It's simple, the number line has no end, and most sequences, like rays, have only a beginning and look like this:

x 1, x 2, x 3,...x n...

Hence the definition of a sequence is a function of the natural argument. In simpler words, this is a series of members of a certain set.

How is the number sequence constructed?

A simple example of a number sequence might look like this: 1, 2, 3, 4, …n…

In most cases, for practical purposes, sequences are built from numbers, and each next member of the series, let's denote it X, has its own name. For example:

x 1 is the first member of the sequence;

x 2 is the second term of the sequence;

x 3 is the third term;

x n is the nth term.

In practical methods, the sequence is given by a general formula in which there is a certain variable. For example:

X n =3n, then the series of numbers itself will look like this:

It is worth remembering that when writing sequences in general, you can use any Latin letters, not just X. For example: y, z, k, etc.

Arithmetic progression as part of sequences

Before looking for the limits of sequences, it is advisable to plunge deeper into the very concept of such a number series, which everyone encountered when they were in middle school. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.

Problem: “Let a 1 = 15, and the progression step of the number series d = 4. Construct the first 4 terms of this series"

Solution: a 1 = 15 (by condition) is the first term of the progression (number series).

and 2 = 15+4=19 is the second term of the progression.

and 3 =19+4=23 is the third term.

and 4 =23+4=27 is the fourth term.

However, using this method it is difficult to reach large values, for example up to a 125. . Especially for such cases, a formula convenient for practice was derived: a n =a 1 +d(n-1). In this case, a 125 =15+4(125-1)=511.

Types of sequences

Most of the sequences are endless, it's worth remembering for the rest of your life. There are two interesting types of number series. The first is given by the formula a n =(-1) n. Mathematicians often call this sequence a flasher. Why? Let's check its number series.

1, 1, -1, 1, -1, 1, etc. With an example like this, it becomes clear that numbers in sequences can easily be repeated.

Factorial sequence. It's easy to guess - the formula defining the sequence contains a factorial. For example: a n = (n+1)!

Then the sequence will look like this:

a 2 = 1x2x3 = 6;

and 3 = 1x2x3x4 = 24, etc.

A sequence defined by an arithmetic progression is called infinitely decreasing if the inequality -1 is satisfied for all its terms

and 3 = - 1/8, etc.

There is even a sequence consisting of the same number. So, n =6 consists of an infinite number of sixes.

Determining the Sequence Limit

Sequence limits have long existed in mathematics. Of course, they deserve their own competent design. So, time to learn the definition of sequence limits. First, let's look at the limit for a linear function in detail:

1. All limits are abbreviated as lim.
2. The notation of a limit consists of the abbreviation lim, any variable tending to a certain number, zero or infinity, as well as the function itself.

It is easy to understand that the definition of the limit of a sequence can be formulated as follows: this is a certain number to which all members of the sequence infinitely approach. A simple example: a x = 4x+1. Then the sequence itself will look like this.

5, 9, 13, 17, 21…x…

Thus, this sequence will increase indefinitely, which means its limit is equal to infinity as x→∞, and it should be written like this:

If we take a similar sequence, but x tends to 1, we get:

And the series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number closer to one (0.1, 0.2, 0.9, 0.986). From this series it is clear that the limit of the function is five.

From this part it is worth remembering what the limit of a numerical sequence is, the definition and method for solving simple problems.

General designation for the limit of sequences

Having examined the limit of a number sequence, its definition and examples, you can proceed to a more complex topic. Absolutely all limits of sequences can be formulated by one formula, which is usually analyzed in the first semester.

So, what does this set of letters, modules and inequality signs mean?

∀ is a universal quantifier, replacing the phrases “for all”, “for everything”, etc.

∃ is an existential quantifier, in this case it means that there is some value N belonging to the set of natural numbers.

A long vertical stick following N means that the given set N is “such that.” In practice, it can mean “such that”, “such that”, etc.

To reinforce the material, read the formula out loud.

Uncertainty and certainty of the limit

The method of finding the limit of sequences, which was discussed above, although simple to use, is not so rational in practice. Try to find the limit for this function:

If we substitute different values ​​of “x” (increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. This results in a rather strange fraction:

But is this really so? Calculating the limit of a number sequence in this case seems quite easy. It would be possible to leave everything as it is, because the answer is ready, and it was received under reasonable conditions, but there is another way specifically for such cases.

First, let's find the highest degree in the numerator of the fraction - this is 1, since x can be represented as x 1.

Now let's find the highest degree in the denominator. Also 1.

Let's divide both the numerator and the denominator by the variable to the highest degree. In this case, divide the fraction by x 1.

Next, we will find what value each term containing a variable tends to. In this case, fractions are considered. As x→∞, the value of each fraction tends to zero. When submitting your work in writing, you should make the following footnotes:

This results in the following expression:

Of course, the fractions containing x did not become zeros! But their value is so small that it is completely permissible not to take it into account in calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero.

What is a neighborhood?

Suppose the professor has at his disposal a complex sequence, given, obviously, by an equally complex formula. The professor has found the answer, but is it right? After all, all people make mistakes.

Auguste Cauchy once came up with an excellent way to prove the limits of sequences. His method was called neighborhood manipulation.

Suppose that there is a certain point a, its neighborhood in both directions on the number line is equal to ε (“epsilon”). Since the last variable is distance, its value is always positive.

Now let's define some sequence x n and assume that the tenth term of the sequence (x 10) is in the neighborhood of a. How can we write this fact in mathematical language?

Let's say x 10 is to the right of point a, then the distance x 10 -a<ε, однако, если расположить «икс десятое» левее точки а, то расстояние получится отрицательным, а это невозможно, значит, следует занести левую часть неравенства под модуль. Получится |х 10 -а|<ε.

Now it’s time to explain in practice the formula discussed above. It is fair to call a certain number a the end point of a sequence if for any of its limits the inequality ε>0 is satisfied, and the entire neighborhood has its own natural number N, such that all members of the sequence with higher numbers will be inside the sequence |x n - a|< ε.

With such knowledge it is easy to solve the sequence limits, prove or disprove the ready-made answer.

Theorems

Theorems on the limits of sequences are an important component of the theory, without which practice is impossible. There are only four main theorems, remembering which can make the solution or proof much easier:

1. Uniqueness of the limit of a sequence. Any sequence can have only one limit or none at all. The same example with a queue that can only have one end.
2. If a series of numbers has a limit, then the sequence of these numbers is limited.
3. The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
4. The limit of the quotient of dividing two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.

Proof of sequences

Sometimes you need to solve an inverse problem, to prove a given limit of a numerical sequence. Let's look at an example.

Prove that the limit of the sequence given by the formula is zero.

According to the rule discussed above, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим:

Let us express n through “epsilon” to show the existence of a certain number and prove the presence of a limit of the sequence.

At this point, it is important to remember that “epsilon” and “en” are positive numbers and are not equal to zero. Now it is possible to continue further transformations using the knowledge about inequalities gained in high school.

How does it turn out that n > -3 + 1/ε. Since it is worth remembering that we are talking about natural numbers, the result can be rounded by putting it in square brackets. Thus, it was proven that for any value of the “epsilon” neighborhood of the point a = 0, a value was found such that the initial inequality is satisfied. From here we can safely say that the number a is the limit of a given sequence. Q.E.D.

This convenient method can be used to prove the limit of a numerical sequence, no matter how complex it may be at first glance. The main thing is not to panic when you see the task.

Or maybe he's not there?

The existence of a consistency limit is not necessary in practice. You can easily come across series of numbers that really have no end. For example, the same “flashing light” x n = (-1) n. it is obvious that a sequence consisting of only two digits, repeated cyclically, cannot have a limit.

The same story is repeated with sequences consisting of one number, fractional ones, having uncertainty of any order during calculations (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculations also occur. Sometimes double-checking your own solution will help you find the sequence limit.

Monotonic sequence

Several examples of sequences and methods for solving them were discussed above, and now let’s try to take a more specific case and call it a “monotonic sequence.”

Definition: any sequence can rightly be called monotonically increasing if the strict inequality x n holds for it< x n +1. Также любую последовательность справедливо называть монотонной убывающей, если для неё выполняется неравенство x n >x n +1.

Along with these two conditions, there are also similar non-strict inequalities. Accordingly, x n ≤ x n +1 (non-decreasing sequence) and x n ≥ x n +1 (non-increasing sequence).

But it’s easier to understand this with examples.

The sequence given by the formula x n = 2+n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.

And if we take x n =1/n, we get the series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing sequence.

Limit of a convergent and bounded sequence

A bounded sequence is a sequence that has a limit. A convergent sequence is a series of numbers that has an infinitesimal limit.

Thus, the limit of a bounded sequence is any real or complex number. Remember that there can only be one limit.

The limit of a convergent sequence is an infinitesimal (real or complex) quantity. If you draw a sequence diagram, then at a certain point it will seem to converge, tend to turn into a certain value. Hence the name - convergent sequence.

Limit of a monotonic sequence

There may or may not be a limit to such a sequence. First, it is useful to understand when it exists; from here you can start when proving the absence of a limit.

Among monotonic sequences, convergent and divergent are distinguished. Convergent is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent is a sequence that has no limit in its set (neither real nor complex).

Moreover, the sequence converges if, in a geometric representation, its upper and lower limits converge.

The limit of a convergent sequence can be zero in many cases, since any infinitesimal sequence has a known limit (zero).

Whatever convergent sequence you take, they are all bounded, but not all bounded sequences converge.

The sum, difference, product of two convergent sequences is also a convergent sequence. However, the quotient can also be convergent if it is defined!

Various actions with limits

Sequence limits are as significant (in most cases) as digits and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits.

First, like digits and numbers, the limits of any sequence can be added and subtracted. Based on the third theorem on the limits of sequences, the following equality holds: the limit of the sum of sequences is equal to the sum of their limits.

Secondly, based on the fourth theorem on the limits of sequences, the following equality is true: the limit of the product of the nth number of sequences is equal to the product of their limits. The same is true for division: the limit of the quotient of two sequences is equal to the quotient of their limits, provided that the limit is not zero. After all, if the limit of sequences is equal to zero, then division by zero will result, which is impossible.

Properties of sequence quantities

It would seem that the limit of the numerical sequence has already been discussed in some detail, but phrases such as “infinitely small” and “infinitely large” numbers are mentioned more than once. Obviously, if there is a sequence 1/x, where x→∞, then such a fraction is infinitesimal, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such quantities have their own characteristics. The properties of the limit of a sequence having any small or large values ​​are as follows:

1. The sum of any number of any number of small quantities will also be a small quantity.
2. The sum of any number of large quantities will be an infinitely large quantity.
3. The product of arbitrarily small quantities is infinitesimal.
4. The product of any number of large numbers is infinitely large.
5. If the original sequence tends to an infinitely large number, then its inverse will be infinitesimal and tend to zero.

In fact, calculating the limit of a sequence is not such a difficult task if you know a simple algorithm. But the limits of consistency is a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution to such expressions. Starting small, you can achieve great heights over time.

For many people, mathematical analysis is just a set of incomprehensible numbers, symbols and definitions, far from real life. However, the world in which we exist is built on numerical patterns, the identification of which helps not only to understand the world around us and solve its complex problems, but also to simplify everyday practical problems. What does a mathematician mean when he says that a number sequence converges? We should talk about this in more detail.

small?

Let's imagine nesting dolls that fit one inside the other. Their sizes, written in the form of numbers, starting with the largest and ending with the smallest of them, form a sequence. If you imagine an infinite number of such bright figures, then the resulting row will turn out to be fantastically long. This is a convergent number sequence. And it tends to zero, since the size of each subsequent nesting doll, catastrophically decreasing, gradually turns into nothing. Thus, it is easy to explain what the infinitesimal is.

A similar example would be a road going into the distance. And the visual dimensions of the car driving away from the observer along it, gradually shrinking, turn into a shapeless speck resembling a point. Thus, the car, like some object, moving away in an unknown direction, becomes infinitely small. The parameters of the specified body will never be zero in the literal sense of the word, but invariably tend to this value in the final limit. Therefore, this sequence converges again to zero.

Let's calculate everything drop by drop

Let us now imagine an everyday situation. The doctor prescribed the patient to take the mixture, starting with ten drops per day and adding two drops every subsequent day. And so the doctor suggested continuing until the contents of the bottle of medicine, the volume of which is 190 drops, are gone. From the above it follows that the number of such, listed by day, will be the following number series: 10, 12, 14 and so on.

How to find out the time to complete the entire course and the number of members of the sequence? Here, of course, you can count the drops in a primitive way. But it is much easier, taking into account the pattern, to use the formula with a step d = 2. And using this method, find out that the number of members of the number series is 10. Moreover, a 10 = 28. The number of the member indicates the number of days of taking the medicine, and 28 corresponds to the number drops that the patient should take on the last day. Does this sequence converge? No, because, despite the fact that it is limited at the bottom by the number 10, and at the top - 28, such a number series has no limit, unlike the previous examples.

What is the difference?

Let us now try to clarify: when a number series turns out to be a convergent sequence. A definition of this kind, as can be concluded from the above, is directly related to the concept of a finite limit, the presence of which reveals the essence of the issue. So what is the fundamental difference between the previously given examples? And why in the last of them the number 28 cannot be considered the limit of the number series X n = 10 + 2(n-1)?

To clarify this question, consider another sequence given by the formula below, where n belongs to the set of natural numbers.

This community of members is a set of ordinary fractions, the numerator of which is 1, and the denominator is constantly increasing: 1, ½ ...

Moreover, each subsequent representative of this series, in terms of location on the number line, approaches 0 more and more. This means that a neighborhood appears where the points cluster around zero, which is the limit. And the closer they are to it, the denser their concentration on the number line becomes. And the distance between them is catastrophically reduced, turning into infinitesimal. This is a sign that the sequence is convergent.

In the same way, the multi-colored rectangles depicted in the figure, when removed in space, are visually arranged more closely together, in the hypothetical limit turning into negligible ones.

Infinitely large sequences

Having examined the definition of a convergent sequence, let us now move on to counterexamples. Many of them have been known to man since ancient times. The simplest variants of divergent sequences are series of natural and even numbers. They are otherwise called infinitely large, since their members, constantly increasing, are increasingly approaching positive infinity.

Examples of these can also be any of the arithmetic and geometric progressions with a step and denominator, respectively, greater than zero. Divergent sequences are also considered to be numerical series that have no limit at all. For example, X n = (-2) n -1 .

Fibonacci sequence

The practical benefits of the previously mentioned number series for humanity are undeniable. But there are many other wonderful examples. One of them is the Fibonacci sequence. Each of its terms, which begin with one, is the sum of the previous ones. Its first two representatives are 1 and 1. The third is 1+1=2, the fourth is 1+2=3, the fifth is 2+3=5. Further, according to the same logic, follow the numbers 8, 13, 21 and so on.

This series of numbers increases indefinitely and has no finite limit. But it has another wonderful property. The ratio of each previous number to the next one is increasingly approaching in its value to 0.618. Here you can understand the difference between a convergent and divergent sequence, because if you compile a series of quotients obtained from divisions, the indicated numerical system will have a final limit equal to 0.618.

Sequence of Fibonacci ratios

The above numerical series is widely used for practical purposes for technical analysis of markets. But this does not limit its capabilities, which the Egyptians and Greeks knew and were able to put into practice in ancient times. This is proven by the pyramids and the Parthenon they built. After all, the number 0.618 is a constant coefficient of the golden ratio, well known in ancient times. According to this rule, any arbitrary segment can be divided so that the relationship between its parts will coincide with the relationship between the largest of the segments and the total length.

Let's build a series of these relationships and try to analyze this sequence. The number series will be as follows: 1; 0.5; 0.67; 0.6; 0.625; 0.615; 0.619 and so on. Continuing in this way, we can verify that the limit of the convergent sequence will indeed be 0.618. However, it is necessary to note other properties of this pattern. Here the numbers seem to be out of order, and not at all in ascending or descending order. This means that this convergent sequence is not monotonic. Why this is so will be discussed further.

Monotony and limitation

Members of a number series with increasing numbers can clearly decrease (if x 1 >x 2 >x 3 >…>x n >…) or increase (if x 1

Having written down the numbers of this series, you can see that any of its members, indefinitely approaching 1, will never exceed this value. In this case, the convergent sequence is said to be bounded. This happens whenever there is a positive number M that always turns out to be greater than any of the terms of the series in modulus. If a number series has signs of monotonicity and has a limit, and therefore converges, then it is necessarily endowed with this property. Moreover, the opposite does not have to be true. This is evidenced by the theorem on the boundedness of a convergent sequence.

The application of such observations in practice turns out to be very useful. Let's give a specific example, examining the properties of the sequence X n = n/n+1, and prove its convergence. It is easy to show that it is monotonic, since (x n +1 - x n) is a positive number for any value of n. The limit of the sequence is equal to the number 1, which means that all the conditions of the above theorem, also called Weierstrass’s theorem, are met. The boundedness theorem for a convergent sequence states that if it has a limit, then it is bounded in any case. However, let's give the following example. The number series X n = (-1) n is bounded below by the number -1 and above by 1. But this sequence is not monotonic, has no limit and therefore does not converge. That is, limitedness does not always imply the presence of a limit and convergence. For this to happen, the lower and upper limits must coincide, as in the case of the Fibonacci ratios.

Numbers and laws of the Universe

The simplest variants of a convergent and divergent sequence are, perhaps, the number series X n = n and X n = 1/n. The first of them is a natural series of numbers. It is, as already mentioned, infinitely large. The second convergent sequence is bounded, and its terms approach infinitesimal in magnitude. Each of these formulas personifies one of the sides of the multifaceted Universe, helping a person, in the language of numbers and signs, to imagine and calculate something unknowable, inaccessible to limited perception.

The laws of the universe, from the insignificant to the incredibly large, are also expressed by the golden coefficient of 0.618. Scientists believe that it lies at the core of the essence of things and is used by nature to form its parts. The previously mentioned relationships between the subsequent and previous members of the Fibonacci series do not complete the demonstration of the amazing properties of this unique series. If we consider the quotient of dividing the previous term by the next one by one, we get the series 0.5; 0.33; 0.4; 0.375; 0.384; 0.380; 0.382 and so on. The interesting thing is that this limited sequence converges, it is not monotonic, but the ratio of adjacent numbers extreme from a certain term always turns out to be approximately equal to 0.382, which can also be used in architecture, technical analysis and other industries.

There are other interesting Fibonacci series coefficients, they all play a special role in nature, and are also used by humans for practical purposes. Mathematicians are confident that the Universe is developing along a kind of “golden spiral” formed from the indicated coefficients. With their help, it is possible to calculate many phenomena occurring on Earth and in space, from the growth of the number of certain bacteria to the movement of distant comets. As it turns out, the DNA code is subject to similar laws.

Decreasing geometric progression

There is a theorem stating the uniqueness of the limit of a convergent sequence. This means that it cannot have two or more limits, which is undoubtedly important for finding its mathematical characteristics.

Let's look at some cases. Any number series made up of members of an arithmetic progression is divergent, except for the case with zero step. The same applies to a geometric progression whose denominator is greater than 1. The limits of such number series are “plus” or “minus” of infinity. If the denominator is less than -1, then there is no limit at all. Other options are also possible.

Let's consider a number series given by the formula X n = (1/4) n -1. At first glance, it is easy to understand that this convergent sequence is bounded because it is strictly decreasing and in no way capable of taking negative values.

Let us write a certain number of its members in a series.

You get: 1; 0.25; 0.0625; 0.015625; 0.00390625 and so on. Quite simple calculations are enough to understand how quickly this geometric progression starts from denominators 0

Fundamental Sequences

Augustin Louis Cauchy, a French scientist, showed the world many works related to mathematical analysis. He gave definitions to such concepts as differential, integral, limit and continuity. He also investigated the basic properties of convergent sequences. In order to understand the essence of his ideas, it is necessary to summarize some important details.

At the very beginning of the article, it was shown that there are such sequences for which there is a neighborhood where the points representing the members of a certain series on the number line begin to crowd together, lining up more and more densely. At the same time, the distance between them decreases as the number of the next representative increases, turning into infinitesimal. Thus, it turns out that in a given neighborhood an infinite number of representatives of a given series are grouped, while outside it there is a finite number of them. Such sequences are called fundamental.

The famous Cauchy criterion, created by a French mathematician, clearly indicates that the presence of such a property is sufficient to prove that the sequence converges. The opposite is also true.

It should be noted that this conclusion of the French mathematician is for the most part of purely theoretical interest. Its application in practice is considered quite difficult, therefore, in order to determine the convergence of series, it is much more important to prove the existence of a finite limit for the sequence. Otherwise, it is considered divergent.

When solving problems, you should also take into account the basic properties of convergent sequences. They are presented below.

Infinite amounts

Such famous ancient scientists as Archimedes, Euclid, Eudoxus used sums of infinite number series to calculate the lengths of curves, volumes of bodies and areas of figures. In particular, this is how it was possible to find out the area of ​​a parabolic segment. For this purpose, the sum of the number series of a geometric progression with q = 1/4 was used. The volumes and areas of other arbitrary figures were found in a similar way. This option was called the “exhaustion” method. The idea was that the body being studied, complex in shape, was divided into parts, which represented figures with easily measurable parameters. For this reason, it was not difficult to calculate their areas and volumes, and then they were added up.

By the way, similar problems are very familiar to modern schoolchildren and are found in Unified State Examination tasks. A unique method, found by distant ancestors, is still the simplest solution today. Even if there are only two or three parts into which a numerical figure is divided, the addition of their areas still represents the sum of the number series.

Much later, the ancient Greek scientists Leibniz and Newton, based on the experience of their wise predecessors, learned the laws of integral calculation. Knowledge of the properties of sequences helped them solve differential and algebraic equations. Currently, the theory of series, created through the efforts of many generations of talented scientists, provides a chance to solve a huge number of mathematical and practical problems. And the study of numerical sequences is the main problem solved by mathematical analysis since its creation.

Sequence is one of the basic concepts of mathematics. The sequence can be made up of numbers, points, functions, vectors, etc. A sequence is considered given if a law is specified according to which each natural number is associated with an element of a certain set. The sequence is written in the form, or briefly. The elements are called members of the sequence, - the first, - the second, - the common (th) member of the sequence.

Number sequences are most often considered, i.e. sequences whose members are numbers. The analytical method is the simplest way to specify a numerical sequence. This is done using a formula expressing the th member of the sequence through its number. For example, if

Another method is recurrent (from the Latin word recurrens - “returning”), when the first few terms of the sequence are specified and a rule that allows you to calculate each subsequent term through the previous ones. For example:

Examples of number sequences are arithmetic progression and geometric progression.

It is interesting to trace the behavior of the members of the sequence as the number increases indefinitely (what increases indefinitely is written in the form and read: “tends to infinity”).

Consider a sequence with a common term: , , , …, , …. All terms of this sequence are different from zero, but the more , the less different from zero. The terms of this sequence tend to zero as they increase indefinitely. They say that the number zero is the limit of this sequence.

Another example: - defines a sequence

The terms of this sequence also tend to zero, but they are sometimes greater than zero, sometimes less than zero - their limit.

Let's look at another example: . If represented in the form

then it will become clear that this sequence tends to unity.

Let us define the limit of a sequence. A number is called the limit of a sequence if for any positive number it is possible to specify a number such that the inequality holds for all.

If there is a limit to the sequence, then they write, or (the first three letters of the Latin word limes - “limit”).

This definition will become clearer if it is given a geometric meaning. Let's enclose the number in an interval (Fig. 1). A number is a limit of a sequence if, regardless of the smallness of the interval, all members of the sequence with numbers greater than some will lie in this interval. In other words, only a finite number of terms of the sequence can be outside any interval.

For the considered sequence, the -neighborhood of the point zero at includes all terms of the sequence except the first ten, and at - all terms of the sequence except the first hundred.

A sequence that has a limit is called convergent, and a sequence that does not have a limit is called divergent. Here is an example of a divergent sequence: . Its members are alternately equal and do not tend to any limit.

If the sequence converges, then it is bounded, i.e. there are numbers and such that all terms of the sequence satisfy the condition. It follows that all unbounded sequences are divergent. These are the sequences:

“A close, deep study of nature is the source of the most fruitful discoveries in mathematics.” J. Fourier

A sequence tending to zero is called infinitesimal. The concept of infinitesimal can be used as the basis for the general definition of the limit of a sequence, since the limit of a sequence is equal if and only if it can be represented as a sum , where is infinitesimal.

The considered sequences are infinitesimal. The sequence , as follows from (2), differs from 1 by infinitesimal , and therefore the limit of this sequence is equal to 1.

The concept of an infinitely large sequence is also of great importance in mathematical analysis. A sequence is called infinitely large if the sequence is infinitesimal. An infinitely large sequence is written in the form , or , and is said to “tend to infinity.” Here are examples of infinitely large sequences:

We emphasize that an infinitely large sequence has no limit.

Let's consider the sequences and . It is possible to define sequences with common terms , , and (if). The following theorem is valid, which is often called the theorem on arithmetic operations with limits: if the sequences are convergent, then the sequences , , , and equalities also hold:

In the latter case, it is necessary to require, in addition to all the terms of the sequence to be different from zero, that the condition be satisfied.

By applying this theorem, many limits can be found. Let us find, for example, the limit of a sequence with a common term and non-increasing ones. It is quite obvious that this sequence tends to some number that is either less than or equal to . In the course of mathematical analysis, the theorem is proven that a non-decreasing and bounded above sequence has a limit (a similar statement is true for a non-increasing and bounded below sequence). This remarkable theorem provides sufficient conditions for the existence of a limit. From it, for example, it follows that the sequence of areas of regular triangles inscribed in a circle of unit radius has a limit, since it is monotonically increasing and bounded from above. The limit of this sequence is indicated by .

Using the limit of a monotonic limited sequence, a number that plays a large role in mathematical analysis is determined - the base of natural logarithms:

.

Sequence (1), as already noted, is monotonic and, moreover, bounded from above. It has a limit. We can easily find this limit. If it is equal, then the number must satisfy the equality. Solving this equation, we get .

Consider a series of natural numbers: 1, 2, 3, , n – 1, n,  .

If we replace every natural number n in this series by a certain number a n, following some law, we get a new series of numbers:

a 1 , a 2 , a 3 , , a n –1 , a n , ,

briefly designated and called numerical sequence. Magnitude a n is called a common member of a number sequence. Usually the number sequence is given by some formula a n = f(n) allowing you to find any member of the sequence by its number n; this formula is called the general term formula. Note that it is not always possible to define a numerical sequence using a general term formula; sometimes a sequence is specified by describing its members.

By definition, a sequence always contains an infinite number of elements: any two different elements differ at least in their numbers, of which there are infinitely many.

A number sequence is a special case of a function. A sequence is a function defined on the set of natural numbers and taking values ​​in the set of real numbers, i.e. a function of the form f : N  R.

Subsequence
called increasing(decreasing), if for any n  N
Such sequences are called strictly monotonous.

Sometimes it is convenient to use not all natural numbers as numbers, but only some of them (for example, natural numbers starting from some natural number n 0). For numbering it is also possible to use not only natural numbers, but also other numbers, for example, n= 0, 1, 2,  (here zero is added as another number to the set of natural numbers). In such cases, when specifying the sequence, indicate what values ​​the numbers take n.

If in some sequence for any n  N
then the sequence is called non-decreasing(non-increasing). Such sequences are called monotonous.

Example 1 . The number sequence 1, 2, 3, 4, 5, ... is a series of natural numbers and has a common term a n = n.

Example 2 . The number sequence 2, 4, 6, 8, 10, ... is a series of even numbers and has a common term a n = 2n.

Example 3 . 1.4, 1.41, 1.414, 1.4142, … – a numerical sequence of approximate values ​​with increasing accuracy.

In the last example it is impossible to give a formula for the general term of the sequence.

Example 4 . Write the first 5 terms of a number sequence using its common term
. To calculate a 1 is needed in the formula for the general term a n instead of n substitute 1 to calculate a 2 − 2, etc. Then we have:

Test 6 . The common member of the sequence 1, 2, 6, 24, 120,  is:

1)

2)

3)

4)

Test 7 .
is:

1)

2)

3)

4)

Test 8 . Common member of the sequence
is:

1)

2)

3)

4)

Number sequence limit

Consider a number sequence whose common term approaches some number A when the serial number increases n. In this case, the number sequence is said to have a limit. This concept has a more strict definition.

Number A called the limit of a number sequence
:

(1)

if for any  > 0 there is such a number n 0 = n 0 (), depending on , which
at n > n 0 .

This definition means that A there is a limit to a number sequence if its common term approaches without limit A with increasing n. Geometrically, this means that for any  > 0 one can find such a number n 0 , which, starting from n > n 0 , all members of the sequence are located inside the interval ( A – , A+ ). A sequence having a limit is called convergent; otherwise - divergent.

A number sequence can have only one limit (finite or infinite) of a certain sign.

Example 5 . Harmonic sequence has the limit number 0. Indeed, for any interval (–; +) as a number N 0 can be any integer greater than . Then for everyone n > n 0 >we have

Example 6 . The sequence 2, 5, 2, 5,  is divergent. Indeed, no interval of length less than, for example, one, can contain all members of the sequence, starting from a certain number.

The sequence is called limited, if such a number exists M, What
for all n. Every convergent sequence is bounded. Every monotonic and bounded sequence has a limit. Every convergent sequence has a unique limit.

Example 7 . Subsequence
is increasing and limited. She has a limit
=e.

Number e called Euler number and approximately equal to 2.718 28.

Test 9 . The sequence 1, 4, 9, 16,  is:

1) convergent;

2) divergent;

3) limited;

Test 10 . Subsequence
is:

1) convergent;

2) divergent;

3) limited;

4) arithmetic progression;

5) geometric progression.

Test 11 . Subsequence is not:

1) convergent;

2) divergent;

3) limited;

4) harmonic.

Test 12 . Limit of a sequence given by a common term
equal.