The limit of the numerical sequence. How to prove that the sequence converges? Basic properties of convergent sequences Types of sequences

Definition of sequence and function limits, properties of limits, first and second remarkable limits, examples.

constant number a called limit sequences(x n) if for any arbitrarily small positive number ε > 0 there exists a number N such that all values x n, for which n>N, satisfy the inequality

Write it as follows: or x n → a.

Inequality (6.1) is equivalent to the double inequality

a - ε< x n < a + ε которое означает, что точки x n, starting from some number n>N, lie inside the interval (a-ε , a+ε), i.e. fall into any small ε-neighborhood of the point a.

A sequence that has a limit is called converging, otherwise - divergent.

The concept of the limit of a function is a generalization of the concept of the limit of a sequence, since the limit of a sequence can be considered as the limit of the function x n = f(n) of an integer argument n.

Let a function f(x) be given and let a - limit point the domain of definition of this function D(f), i.e. such a point, any neighborhood of which contains points of the set D(f) different from a. Dot a may or may not belong to the set D(f).

Definition 1. The constant number A is called limit functions f(x) at x→ a if for any sequence (x n ) of argument values ​​tending to a, the corresponding sequences (f(x n)) have the same limit A.

This definition is called defining the limit of a function according to Heine, or " in the language of sequences”.

Definition 2. The constant number A is called limit functions f(x) at x→a if, given an arbitrary, arbitrarily small positive number ε, one can find δ >0 (depending on ε) such that for all x, lying in the ε-neighborhood of the number a, i.e. for x satisfying the inequality
0 < x-a < ε , значения функции f(x) будут лежать в ε-окрестности числа А, т.е. |f(x)-A| < ε

This definition is called defining the limit of a function according to Cauchy, or “in the language ε - δ"

Definitions 1 and 2 are equivalent. If the function f(x) as x → a has limit equal to A, this is written as

In the event that the sequence (f(x n)) increases (or decreases) indefinitely for any method of approximation x to your limit a, then we will say that the function f(x) has infinite limit, and write it as:

A variable (i.e. a sequence or function) whose limit is zero is called infinitely small.

A variable whose limit is equal to infinity is called infinitely large.

To find the limit in practice, use the following theorems.

Theorem 1 . If every limit exists

(6.4)

(6.5)

(6.6)

Comment. Expressions of the form 0/0, ∞/∞, ∞-∞ 0*∞ are indefinite, for example, the ratio of two infinitesimal or infinitely large quantities, and finding a limit of this kind is called “uncertainty disclosure”.

Theorem 2.

those. it is possible to pass to the limit at the base of the degree at a constant exponent, in particular,

Theorem 3.

(6.11)

where e» 2.7 is the base of the natural logarithm. Formulas (6.10) and (6.11) are called the first remarkable limit and the second remarkable limit.

The corollaries of formula (6.11) are also used in practice:

(6.12)

(6.13)

(6.14)

in particular the limit

If x → a and at the same time x > a, then write x →a + 0. If, in particular, a = 0, then write +0 instead of the symbol 0+0. Similarly, if x→a and at the same time x and are named accordingly. right limit and left limit functions f(x) at the point a. For the limit of the function f(x) to exist as x→ a, it is necessary and sufficient that . The function f(x) is called continuous at the point x 0 if limit

(6.15)

Condition (6.15) can be rewritten as:

that is, passage to the limit under the sign of a function is possible if it is continuous at a given point.

If equality (6.15) is violated, then we say that at x = xo function f(x) It has gap. Consider the function y = 1/x. The domain of this function is the set R, except for x = 0. The point x = 0 is a limit point of the set D(f), since in any of its neighborhoods, i.e., any open interval containing the point 0 contains points from D(f), but it does not itself belong to this set. The value f(x o)= f(0) is not defined, so the function has a discontinuity at the point x o = 0.

The function f(x) is called continuous on the right at a point x o if limit

and continuous on the left at a point x o if limit

Continuity of a function at a point x o is equivalent to its continuity at this point both on the right and on the left.

For a function to be continuous at a point x o, for example, on the right, it is necessary, firstly, that there is a finite limit , and secondly, that this limit be equal to f(x o). Therefore, if at least one of these two conditions is not met, then the function will have a gap.

1. If the limit exists and is not equal to f(x o), then they say that function f(x) at the point xo has break of the first kind, or jump.

2. If the limit is +∞ or -∞ or does not exist, then they say that in point x o the function has a break second kind.

For example, the function y = ctg x as x → +0 has a limit equal to +∞ , which means that at the point x=0 it has a discontinuity of the second kind. Function y = E(x) (integer part of x) at points with integer abscissas has discontinuities of the first kind, or jumps.

A function that is continuous at every point of the interval is called continuous in . A continuous function is represented by a solid curve.

Many problems associated with the continuous growth of some quantity lead to the second remarkable limit. Such tasks, for example, include: the growth of the contribution according to the law of compound interest, the growth of the country's population, the decay of a radioactive substance, the multiplication of bacteria, etc.

Consider example of Ya. I. Perelman, which gives the interpretation of the number e in the compound interest problem. Number e there is a limit . In savings banks, interest money is added to the fixed capital annually. If the connection is made more often, then the capital grows faster, since a large amount is involved in the formation of interest. Let's take a purely theoretical, highly simplified example. Let the bank put 100 den. units at the rate of 100% per annum. If interest-bearing money is added to the fixed capital only after a year, then by this time 100 den. units will turn into 200 den. Now let's see what 100 den will turn into. units, if interest money is added to the fixed capital every six months. After half a year 100 den. units will grow by 100 × 1.5 = 150, and in another six months - by 150 × 1.5 = 225 (money units). If the accession is done every 1/3 of the year, then after a year 100 den. units will turn into 100 × (1 + 1/3) 3 ≈ 237 (den. units). We will increase the timeframe for adding interest money to 0.1 year, 0.01 year, 0.001 year, and so on. Then out of 100 den. units a year later:

100×(1 +1/10) 10 ≈ 259 (den. units),

100×(1+1/100) 100 ≈ 270 (den. units),

100×(1+1/1000) 1000 ≈271 (den. units).

With an unlimited reduction in the terms of joining interest, the accumulated capital does not grow indefinitely, but approaches a certain limit equal to approximately 271. The capital placed at 100% per annum cannot increase more than 2.71 times, even if the accrued interest were added to the capital every second because the limit

Example 3.1. Using the definition of the limit of a number sequence, prove that the sequence x n =(n-1)/n has a limit equal to 1.

Solution. We need to prove that whatever ε > 0 we take, there is a natural number N for it, such that for all n > N the inequality |x n -1|< ε

Take any ε > 0. Since x n -1 =(n+1)/n - 1= 1/n, then to find N it is enough to solve the inequality 1/n<ε. Отсюда n>1/ε and, therefore, N can be taken as the integer part of 1/ε N = E(1/ε). We thus proved that the limit .

Solution. Apply the limit sum theorem and find the limit of each term. As n → ∞, the numerator and denominator of each term tends to infinity, and we cannot apply the quotient limit theorem directly. Therefore, we first transform x n, dividing the numerator and denominator of the first term by n 2, and the second n. Then, applying the quotient limit theorem and the sum limit theorem, we find:

Example 3.3. . Find .

Here we have used the degree limit theorem: the limit of a degree is equal to the degree of the limit of the base.

Example 3.4. Find ( ).

Solution. It is impossible to apply the difference limit theorem, since we have an uncertainty of the form ∞-∞. Let's transform the formula of the general term:

Example 3.5. Given a function f(x)=2 1/x . Prove that the limit does not exist.

Solution. We use the definition 1 of the limit of a function in terms of a sequence. Take a sequence ( x n ) converging to 0, i.e. Let us show that the value f(x n)= behaves differently for different sequences. Let x n = 1/n. Obviously, then the limit Let's choose now as x n a sequence with a common term x n = -1/n, also tending to zero. Therefore, there is no limit.

Example 3.6. Prove that the limit does not exist.

Solution. Let x 1 , x 2 ,..., x n ,... be a sequence for which
. How does the sequence (f(x n)) = (sin x n ) behave for different x n → ∞

If x n \u003d p n, then sin x n \u003d sin (p n) = 0 for all n and limit If
xn=2 p n+ p /2, then sin x n = sin(2 p n+ p /2) = sin p /2 = 1 for all n and hence the limit. Thus does not exist.

Number sequences are infinite sets of numbers. Examples of sequences are: the sequence of all members of an infinite geometric progression, the sequence of approximate values ​​( x 1 = 1, x 2 = 1,4, x 3= 1.41, ...), the sequence of perimeters of regular n-gons inscribed in a given circle. Let us refine the notion of a numerical sequence.

Definition 1. If every number n from the natural series of numbers 1, 2, 3,..., P,... assigned a real number x p, then the set of real numbers

x 1 , x 2 , x 3 , …, x n , …(2.1)

called number sequence, or just a sequence. .

Numbers x 1 , x 2, x 3, ..., x p,... will call elements, or members sequences (2.1), symbol x p - general an element, or a member of a sequence, and the number P - his number. Briefly, the sequence (2.1) will be denoted by the symbol (x p ). For example, the character (1/ n) denotes a sequence of numbers

In other words, a sequence can be understood as an infinite set of numbered elements or a set of pairs of numbers (p, x p), in which the first number takes the consecutive values ​​1, 2, 3, ... . A sequence is considered given if a method for obtaining any of its elements is specified. For example, the formula x n = -1 + (-1)n defines the sequence 0, 2, 0, 2,... .

Geometrically, the sequence is depicted on the numerical axis as a sequence of points whose coordinates are equal to the corresponding members of the sequence. On fig. 2.1 shows the sequence ( x n} = {1/n) on the number line.

The concept of a convergent sequence

Definition 2. Number a called sequence limit{x n} , if for any positive number ε there is a number N, that for all n > N the inequality

A sequence that has a limit is called converging. If the sequence has a number as its limit a, then it is written like this:

A sequence that has no limit is called divergent.

Definition 3. A sequence that has a number as its limit a= 0 is called infinitesimal sequence.

Remark 1. Let the sequence ( x n) has as its limit the number a. Then the sequence (α n} = {x n - a) is infinitely small, i.e. any element x p convergent sequence with limit a, can be represented as

where α n- element of an infinitesimal sequence (α n} .

Remark 2. Inequality (2.2) is equivalent to inequalities (see property 4 of the modulus of a number from § 1.5)

This means that at n > N all elements of the sequence ( x n) are situated in ε-neighbourhood points a(Fig. 2.2), and the number N is determined by the value of ε.

It is interesting to give a geometric interpretation of this definition. Since the sequence is an infinite set of numbers, then if it converges, in any ε-neighborhood of the point a on the real line there is an infinite number of points - elements of this sequence, while outside the ε-neighbourhood there is a finite number of elements. Therefore, the limit of a sequence is often called thickening point.

Remark 3. Unlimited sequence has no final limit. However, she may have endless limit, which is written in the following form:

If at the same time, starting from a certain number, all members of the sequence are positive (negative), then write

If a ( x n) is an infinitesimal sequence, then (1 /x p} - an infinite sequence which has an infinite limit in the sense of (2.3), and vice versa.

Let us give examples of convergent and divergent sequences.

Example 1 Show, using the definition of the limit of a sequence, that .

Solution. Take any number ε > 0. Since

then for inequality (2.2) to hold, it suffices to solve the inequality 1 / ( n + 1) < ε, откуда получаем n> (1 - ε) / ε. Enough to take N= [(1 - ε)/ε] (the integer part of the number (1 - ε)/ ε)* so that the inequality |x p - 1| < ε выполнялосьпривсех n > N.

* Symbol [ a] means the integer part of the number a, i.e. largest integer not exceeding a. For example, =2, =2, =0, [-0, 5] = -1, [-23.7] = -24.

Example 2 Show that the sequence ( x n} = (-1)n, or -1, 1, -1, 1,... has no limit.

Solution. Indeed, whatever number we assume as a limit: 1 or -1, with ε< 0,5 неравенство (2.2), определяющее предел последовательности, не удовлетво­ряется - вне ε -окрестности этих чисел остается бесконечное число элементов x p: all odd-numbered elements are -1, even-numbered elements are 1.

Basic properties of convergent sequences

Let us present the main properties of convergent sequences, which are formulated in the form of theorems in the course of higher mathematics.

1.If all elements of an infinitesimal sequence{x n} are equal to the same number c, then c = 0.

2. A convergent sequence has only one limit.

3.The convergent sequence is bounded.

4.Sum (difference) of convergent sequences{x n} and{y n} is a convergent sequence whose limit is equal to the sum (difference) of the limits of the sequences{x p} and{y p}.

5.Product of convergent sequences{x n} and{y n} is a convergent sequence whose limit is equal to the product of the limits of the sequences{x n} and{y n} .

6.Quotient of two convergent sequences{x n} and{y n} provided that the limit of the sequence{y n} is nonzero, there is a convergent sequence whose limit is equal to the quotient of the limits of the sequences{x n} and{y p} .

7. If the elements of a convergent sequence{x n} satisfy the inequality x p ≥ b (x p ≤ b) starting from some number, then the limit a of this sequence also satisfies the inequality a ≥ b (a ≤ b).

8.The product of an infinitesimal sequence by a bounded sequence or by a number is an infinitesimal sequence.

9.The product of a finite number of infinitesimal sequences is an infinitesimal sequence.

Let's consider the application of these properties with examples.

Example 3. Find the limit.

Solution. At n the numerator and denominator of the fraction tend to infinity, i.e. the quotient limit theorem cannot be applied immediately, since it assumes the existence of finite limits of sequences. We transform this sequence by dividing the numerator and denominator by n 2. Applying then the theorems on the limit of the quotient, the limit of the sum, and again the limit of the quotient, we successively find

Example 4 x p) = at P.

Solution. Here, as in the previous example, the numerator and denominator have no finite limits, and therefore the appropriate transformations must first be performed. Dividing the numerator and denominator by n, we get

Since the numerator contains the product of an infinitesimal sequence and a bounded sequence, then, by property 8, we finally obtain

Example 5 Find the limit of the sequence ( x n) = at P .

Solution. Here it is impossible to apply directly the theorem on the limit of the sum (difference) of sequences, since there are no finite limits of the terms in the formula for ( x n} . Multiply and divide the formula for ( x n) to the conjugate expression :

Number e

Consider the sequence ( x n} , whose common term is expressed by the formula

In the course of mathematical analysis, it is proved that this sequence increases monotonically and has a limit. This limit is called the number e. Therefore, by definition

Number e plays a big role in mathematics. Next, a method for calculating it with any required accuracy will be considered. Note here that the number e is irrational; its approximate value is e = 2,7182818... .

3. Limit of number sequence

3.1. The concept of a numerical sequence and a function of a natural argument

Definition 3.1. A numerical sequence (hereinafter simply a sequence) is an ordered countable set of numbers

{x1, x2, x3, ... }.

Pay attention to two points.

1. There are infinitely many numbers in the sequence. If there are a finite number of numbers, this is not a sequence!

2. All numbers are ordered, that is, arranged in a certain order.

In what follows, we will often use the abbreviation for the sequence ( xn}.

Certain operations can be performed on sequences. Let's consider some of them.

1. Multiplication of a sequence by a number.

Subsequence c×{ xn) is a sequence with elements ( c× xn), that is

c×{ x1, x2, x3, ... }={c× x1, s× x2, s× x3, ... }.

2. Addition and subtraction of sequences.

{xn}±{ yn}={xn± yn},

or, in more detail,

{x1, x2, x3, ...}±{ y1, y2, y3, ... }={x1± y1, x2± y2, x3± y3, ... }.

3. Multiplication of sequences.

{xn}×{ yn}={xn× yn}.

4. Division of sequences.

{xn}/{yn}={xn/yn}.

Naturally, it is assumed that in this case all yn¹ 0.

Definition 3.2. Subsequence ( xn) is called bounded from above if https://pandia.ru/text/78/243/images/image004_49.gif" width="71 height=20" height="20">.gif" width="53" height= "25 src=">. A sequence (xn) is called bounded if it is bounded both above and below.

3.2. Sequence limit. Infinitely large sequence

Definition 3.3. Number a is called the limit of the sequence ( xn) at n tending to infinity, if

https://pandia.ru/text/78/243/images/image007_38.gif" width="77" height="33 src=">.gif" width="93" height="33"> if .

They say that if .

Definition 3.4. Subsequence ( xn) is called infinitely large if (that is, if ).

3.3. An infinitesimal sequence.

Definition 3.5. A sequence (xn) is called infinitesimal if , that is, if .

Infinitesimal sequences have the following properties.

1. The sum and difference of infinitesimal sequences is also an infinitesimal sequence.

2. An infinitesimal sequence is bounded.

3. The product of an infinitesimal sequence and a bounded sequence is an infinitesimal sequence.

4. If ( xn) is an infinitely large sequence, then starting from some N, the sequence (1/ xn), and it is an infinitesimal sequence. Conversely, if ( xn) is an infinitesimal sequence and all xn are different from zero, then (1/ xn) is an infinitely large sequence.

3.4. convergent sequences.

Definition 3.6. If there is an end limit https://pandia.ru/text/78/243/images/image017_29.gif" width="149" height="33">.

5. If , then .

3.5. Passage to the limit in inequalities.

Theorem 3.1. If, starting from some N, all xn ³ b, then .

Consequence. If, starting from some N, all xn ³ yn, then .

Comment. Note that if, starting from some N, all xn > b, then , that is, when passing to the limit, the strict inequality can become non-strict.

Theorem 3.2.("Theorem of two policemen") If, starting from some N, the following properties hold

1..gif" width="163" height="33 src=">,

then exists.

3.6. The limit of a monotone sequence.

Definition 3.7. Subsequence ( xn) is called monotonically increasing if for any n xn+1 ³ xn.

Subsequence ( xn) is called strictly monotonically increasing if for any n xn+1> xn.

xn­.

Definition 3.8. Subsequence ( xn) is called monotonically decreasing if for any n xn+1 £ xn.

Subsequence ( xn) is called strictly monotonically decreasing if for any n xn+1< xn.

Both of these cases are combined with the symbol xn¯.

Theorem on the existence of a limit of a monotone sequence.

1. If the sequence ( xn) is monotonically increasing (decreasing) and bounded from above (from below), then it has a finite limit equal to sup( xn) (inf( xn}).

2 If the sequence ( xn) monotonically increases (decreases), but is not limited from above (from below), then it has a limit equal to +¥ (-¥).

Based on this theorem, it is proved that there is a so-called remarkable limit

https://pandia.ru/text/78/243/images/image028_15.gif" width="176" height="28 src=">. It is called a sequence subsequence ( xn}.

Theorem 3.3. If the sequence ( xn) converges and its limit is a, then any of its subsequences also converges and has the same limit.

If a ( xn) is an infinitely large sequence, then any of its subsequences is also infinitely large.

Bolzano-Weierstrass lemma.

1. From any bounded sequence, one can extract a subsequence that converges to a finite limit.

2. An infinitely large subsequence can be extracted from any unbounded sequence.

On the basis of this lemma, one of the main results of the theory of limits is proved - Bolzano-Cauchy convergence criterion.

In order for the sequence ( xn) there was a finite limit, it is necessary and sufficient that

A sequence that satisfies this property is called a fundamental sequence, or a sequence that converges in itself.

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

small?

Imagine matryoshka 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 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 is infinitely small.

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 dot. Thus, the machine, like an 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's imagine a real life situation. The doctor prescribed the patient to take the medicine, starting with ten drops a day and adding two every next day. And so the doctor suggested continuing until the contents of the vial of medicine, the volume of which is 190 drops, run out. From the foregoing, it follows that the number of such, painted 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, given the pattern, to use the formula with a step of d = 2. And using this method, find out that the number of members of the number series is 10. In this case, a 10 = 28. The member number indicates the number of days of taking the medicine, and 28 corresponds to the number drops that the patient should use on the last day. Does this sequence converge? No, because, despite the fact that it is limited to 10 from below and 28 from above, such a number series has no limit, unlike the previous examples.

What is the difference?

Now let's try to clarify: when the 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 issue, 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, is increasingly approaching 0. This means that such 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 an infinitesimal one. This is a sign that the sequence is converging.

Similarly, the multi-colored rectangles shown in the figure, when moving away in space, are visually more crowded, in the hypothetical limit turning into negligible.

Infinitely large sequences

Having analyzed the definition of a convergent sequence, we now turn to counterexamples. Many of them have been known to man since ancient times. The simplest variants of divergent sequences are the series of natural and even numbers. They are called infinitely large in another way, since their members, constantly increasing, are increasingly approaching positive infinity.

Any of the arithmetic and geometric progressions with a step and a denominator greater than zero, respectively, can also serve as examples of such. Divergent sequences are considered, moreover, numerical series, which do not have a limit at all. For example, X n = (-2) n -1 .

Fibonacci sequence

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

This series of numbers grows indefinitely and has no finite limit. But it has another wonderful property. The ratio of each previous number to the next is more and more close in its value to 0.618. Here you can understand the difference between a convergent and divergent sequence, because if you make a series of received private divisions, the specified numerical system will have a final limit equal to 0.618.

Fibonacci Ratio Sequence

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

Let's build a series of these relations 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, one can verify that the limit of the convergent sequence will indeed be 0.618. However, it is necessary to note other properties of this regularity. Here the numbers seem to go randomly, and not at all in ascending or descending order. This means that this convergent sequence is not monotone. Why this is so will be discussed further.

monotony and limitation

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

Having painted the numbers of this series, one can notice that any of its members, approaching 1 indefinitely, will never exceed this value. In this case, the convergent sequence is said to be bounded. This happens whenever there is such a positive number M, which is always greater than any of the terms of the series modulo. If a number series has signs of monotonicity and has a limit, and therefore converges, then it is necessarily endowed with such a property. And the opposite doesn't have to be true. This is evidenced by the boundedness theorem for a convergent sequence.

The application of such observations in practice turns out to be very useful. Let's give a specific example by 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 values ​​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 the Weierstrass theorem, are satisfied. The theorem on the boundedness of a convergent sequence states that if it has a limit, then in any case it turns out to be bounded. However, let's take the following example. The number series X n = (-1) n is bounded from below by -1 and from above by 1. But this sequence is not monotone, has no limit, and therefore does not converge. That is, the existence of a limit and convergence does not always follow from limitation. For this to work, the lower and upper limits must match, as in the case of Fibonacci ratios.

Numbers and laws of the universe

The simplest variants of a convergent and divergent sequence are, perhaps, the numerical 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 are close to infinitesimal in magnitude. Each of these formulas personifies one of the sides of the multifaceted Universe, helping a person to imagine and calculate something unknowable, inaccessible to limited perception in the language of numbers and signs.

The laws of the universe, ranging from negligible to incredibly large, are also expressed by the golden ratio of 0.618. Scientists believe that it is the basis of the essence of things and is used by nature to form its parts. The relations between the next and the previous members of the Fibonacci series, which we have already mentioned, 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 through one, then we get a series of 0.5; 0.33; 0.4; 0.375; 0.384; 0.380; 0.382 and so on. It is interesting that this limited sequence converges, it is not monotonous, but the ratio of the neighboring numbers extreme from a certain member always approximately equals 0.382, which can also be used in architecture, technical analysis and other industries.

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

Decreasing geometric progression

There is a theorem asserting 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 consider some cases. Any numerical series composed of members of an arithmetic progression is divergent, except for the case with a zero step. The same applies to a geometric progression, the denominator of which is greater than 1. The limits of such numerical series are the "plus" or "minus" of infinity. If the denominator is less than -1, then there is no limit at all. Other options are also possible.

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

Let's write some number of its members in a row.

Get: 1; 0.25; 0.0625; 0.015625; 0.00390625 and so on. Quite simple calculations are enough to understand how fast a given geometric progression with denominators 0

Fundamental sequences

Augustin Louis Cauchy, a French scientist, revealed to the world many works related to mathematical analysis. He gave definitions to such concepts as differential, integral, limit, and continuity. He also studied 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 real line begin to cluster, 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 an infinitely small one. Thus, it turns out that in a given neighborhood an infinite number of representatives of a given series are grouped, while outside of it there are 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 reverse is also true.

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

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

Infinite sums

Such famous scientists of antiquity as Archimedes, Euclid, Eudoxus used the sums of infinite number series to calculate the lengths of curves, volumes of bodies and areas of figures. In particular, in this way it was possible to find out the area of ​​the parabolic segment. For this, the sum of the numerical 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 studied body, complex in shape, was broken into parts, which were figures with easily measured parameters. For this reason, it was not difficult to calculate their areas and volumes, and then they were added together.

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

Much later than the ancient Greek scientists Leibniz and Newton, based on the experience of their wise predecessors, they learned the laws of integral calculation. Knowledge of the properties of sequences helped them solve differential and algebraic equations. At present, the theory of series, created by the efforts of many generations of talented scientists, gives 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 inception.

Sequence is one of the basic concepts of mathematics. The sequence can be composed of numbers, points, functions, vectors, and so on. A sequence is considered given if a law is specified according to which each natural number n is associated with an element x n of some set. The sequence is written as x 1 , x 2 , …, x n , or briefly (x n). Elements x 1 , x 2 , ..., x n are called members of the sequence, x 1 - the first, x 2 - the second, x n - common (n-th) member of the sequence.

Most often, numerical sequences are considered, that is, sequences whose members are numbers. The analytical method is the simplest way to specify a numerical sequence. This is done using a formula that expresses the nth member of the sequence x 1 in terms of its number n. For example, if

Another way is recurrent (from the Latin word recurrences- “returning”), when the first few members of the sequence and the rule are set, allowing each next member to be calculated 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 n increases without limit (the fact that n increases indefinitely is written as n → ∞ and reads: “n tends to infinity”).

Consider a sequence with a common term x n = 1/n: x 1 = 1, x 2 = 1/2; x 3 \u003d 1/3, ..., x 100 \u003d 1/100, .... All members of this sequence are non-zero, but the larger n, the less x n differs from zero. The terms of this sequence tend to zero as n increases indefinitely. The number zero is said to be the limit of this sequence.

Another example: x n = (−1) n / n - defines the sequence

The members of this sequence also tend to zero, but they are either greater than zero or less than zero - their limit.

Consider another example: x n = (n − 1)/(n + 1). If we represent x n in the form

then it becomes clear that this sequence tends to unity.

Let us define the limit of a sequence. A number a is called the limit of a sequence (x n) if, for any positive number ε, one can specify a number N such that, for all n > N, the inequality |x n − a|< ε.

If a is the limit of the sequence (x n), then write x n → a, or a = lim n→∞ x n (lim are the first three letters of the Latin word limes- "limit").

This definition will become clearer if we give it a geometric meaning. We enclose the number a in the interval (a − ε, a + ε) (see the figure). The number a is the limit of the sequence (x n) if, regardless of the smallness of the interval (a − ε, a + ε), all members of the sequence with numbers greater than some N lie in this interval. In other words, outside any interval (a − ε, a + ε) there can be only a finite number of members of the sequence.

For the considered sequence x n = (−1) n /n, the ε-neighborhood of the zero point at ε = 1/10 includes all members of the sequence, except for the first ten, and for ε = 1/100, all the members of the sequence, except for 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: x n = (−1) n . Its terms are alternately +1 and −1 and do not tend to any limit.

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

A sequence tending to zero is said to be 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 (x n) is equal to a if and only if x n can be represented as a sum x n = a + α n , where α n is infinitesimal.

The considered sequences (1/n), ((−1) n /n) are infinitesimal. The sequence (n − 1)/(n + 1), as follows from (2), differs from 1 by an infinitesimal 2/(n + 1), and therefore the limit of this sequence is 1.

Of great importance in mathematical analysis is also the concept of an infinitely large sequence. A sequence (x n) is called infinitely large if the sequence (1/x n) is infinitely small. An infinitely large sequence (x n) is written as x n → ∞, or lim n→∞ x n = ∞, and is said to "go to infinity". Here are examples of infinitely large sequences:

(n 2), (2 n), (√(n + 1)), (n - n 2).

We emphasize that an infinitely large sequence has no limit.

Consider the sequences (x n) and (y n). You can define sequences with common terms x n + y n , x n − y n , x n y n and (if y n ≠ 0) x n /y n . The following theorem is true, which is often called the theorem on arithmetic operations with limits: if the sequences (x n) and (y n) converge, then the sequences (x n + y n), (x n − y n), (x n y n), (x n /y n) also converge and the following equalities hold:

In the latter case, it is necessary to require, in addition, that all members of the sequence (y n) be nonzero, and also that the condition lim n→∞ y n ≠ 0 be satisfied.

By applying this theorem, many limits can be found. Find, for example, the limit of a sequence with a common term

Representing x n in the form

establish that the limit of the numerator and denominator exists:

so we get:

lim n→∞ x n = 2/1 =2.

An important class of sequences is monotone sequences. So called sequences increasing (x n+1 > x n for any n), decreasing (x n+1< x n), неубывающие (x n+1 ≥ x n) и невозрастающие (x n+1 ≤ x n). Последовательность (n − 1)/(n + 1) возрастающая, последовательность (1/n) убывающая. Можно доказать, что рекуррентно заданная последовательность (1) монотонно возрастает.

Imagine that the sequence (x n) does not decrease, i.e., the inequalities

x 1 ≤ x 2 ≤ x 3 ≤ … ≤ x n ≤ x n+1 ≤ …,

and let, in addition, this sequence be bounded from above, i.e., all x n do not exceed some number d. Each member of such a sequence is greater than or equal to the previous one, but none of them exceeds d. It is quite obvious that this sequence tends to some number that is either less than d or equal to d. In the course of mathematical analysis, a theorem is proved that a non-decreasing and bounded from above sequence has a limit (a similar statement is true for a non-increasing and bounded from below sequence). This remarkable theorem gives sufficient conditions for the existence of a limit. From it, for example, it follows that the sequence of areas of regular n-gons 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 denoted by π.

Using the limit of a monotone bounded sequence, the number e, which plays a large role in mathematical analysis, is determined - the base of natural logarithms:

e = lim n→∞ (1 + 1/n) n .

Sequence (1), as already noted, is monotone and, moreover, bounded from above. She has a limit. We can easily find this limit. If it is equal to a, then the number a must satisfy the equality a = √(2 + a). Solving this equation, we get a = 2.