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. The 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
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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.
Mathematics is the science that builds the world. Both the scientist and the common man - no one can do without it. First, young children are taught to count, then add, subtract, multiply, and divide, to high school letter designations come into play, and in the older one you can no longer do without them.
But today we will talk about what all known mathematics is based on. About the 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 such a construction 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 to the store, this is one sequence. And if one person suddenly leaves this queue, then this is a different queue, a different order.
The word "limit" is also easily interpreted - this is the end of something. However, in mathematics, the limits of sequences are those values on the number line that a sequence of numbers tends to. Why strives and does 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. More in simple terms is a series of members of some set.
How is a number sequence built?
The simplest 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 by X, has its own name. For example:
x 1 - the first member of the sequence;
x 2 - the second member of the sequence;
x 3 - the third member;
x n is the nth member.
AT practical methods the sequence is given general formula, which contains some variable. For example:
X n \u003d 3n, then the series of numbers itself will look like this:
It is worth remembering that in the general notation of sequences, you can use any Latin letters, and 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 delve deeper into the very concept of such a number series, which everyone encountered when they were in the middle classes. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.
Task: “Let a 1 \u003d 15, and the step of the progression of the number series d \u003d 4. Build the first 4 members of this row"
Solution: a 1 = 15 (by condition) is the first member of the progression (number series).
and 2 = 15+4=19 is the second member of the progression.
and 3 \u003d 19 + 4 \u003d 23 is the third term.
and 4 \u003d 23 + 4 \u003d 27 is the fourth term.
However, with 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 \u003d a 1 + d (n-1). In this case, a 125 \u003d 15 + 4 (125-1) \u003d 511.
Sequence types
Most of the sequences are endless, it's worth remembering for a lifetime. There are two interesting species number line. The first is given by the formula a n =(-1) n . Mathematicians often refer to this flasher sequences. Why? Let's check its numbers.
1, 1, -1 , 1, -1, 1, etc. With this example, it becomes clear that numbers in sequences can easily be repeated.
factorial sequence. It is easy to guess that there is a factorial in the formula that defines the sequence. For example: and n = (n+1)!
Then the sequence will look like this:
and 2 \u003d 1x2x3 \u003d 6;
and 3 \u003d 1x2x3x4 \u003d 24, etc.
A sequence given by an arithmetic progression is called infinitely decreasing if the inequality -1 is observed for all its members and 3 \u003d - 1/8, etc. There is even a sequence consisting of the same number. So, and n \u003d 6 consists of an infinite number of sixes. 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, consider the limit for a linear function in detail: It is easy to understand that the definition of the limit of a sequence can be formulated as follows: it is a certain number, to which all members of the sequence infinitely approach. Simple example: and 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 that its limit is equal to infinity as x→∞, and this should be written as follows: 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 more and more close to one (0.1, 0.2, 0.9, 0.986). It can be seen from this series 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 tasks. Having analyzed the limit of the numerical sequence, its definition and examples, we 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 existence 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 consolidate the material, read the formula aloud. 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 x values (increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. It turns out a rather strange fraction: But is it really so? Calculating the limit of the numerical sequence in this case seems easy enough. It would be possible to leave everything as it is, because the answer is ready, and it was received on reasonable terms, 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. Divide both the numerator and the denominator by the variable to the highest degree. In this case, we divide the fraction by x 1. Next, let's find what value each term containing the variable tends to. In this case, fractions are considered. As x→∞, the value of each of the fractions tends to zero. When making a paper in writing, it is worth making the following footnotes: The following expression is obtained: Of course, the fractions containing x did not become zeros! But their value is so small that it is quite permissible not to take it into account in the calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero. Let us assume that the professor has at his disposal a complex sequence, given, obviously, by a no less complex formula. The professor found the answer, but does it fit? After all, all people make mistakes. Auguste Cauchy came up with a great way to prove the limits of sequences. His method was called neighborhood operation. Suppose that there is some point a, its neighborhood in both directions on the real line is equal to ε ("epsilon"). Since the last variable is distance, its value is always positive. Now let's set some sequence x n and suppose that the tenth member of the sequence (x 10) is included in the neighborhood of a. How to write this fact in mathematical language? Suppose 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 is time to explain in practice the formula mentioned above. It is fair to call a certain number a the end point of a sequence if the inequality ε>0 holds for any of its limits, 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 limits of a sequence, to prove or disprove a ready answer. 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, you can significantly facilitate the process of solving or proving: Sometimes it is required 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 equal to zero. According to the above rule, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим: Let's express n in terms of "epsilon" to show the existence of a certain number and prove the existence of a sequence limit. At this stage, it is important to recall that "epsilon" and "en" are positive numbers and not equal to zero. Now you can continue further transformations using the knowledge about inequalities gained in high school. Whence it turns 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 proved 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 this we can safely assert that the number a is the limit of the given sequence. Q.E.D. With such a convenient method, you can prove the limit of a numerical sequence, no matter how complicated it may seem at first glance. The main thing is not to panic at the sight of the task. The existence of a sequence limit is not necessary in practice. It is easy to find such series of numbers that really have no end. For example, the same flasher x n = (-1) n . it is obvious that a sequence consisting of only two digits cyclically repeating cannot have a limit. The same story is repeated with sequences consisting of a single number, fractional, having in the course of calculations an uncertainty of any order (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculation also takes place. Sometimes rechecking your own solution will help you find the limit of successions. Above, we considered several examples of sequences, methods for solving them, and now let's try to take a more specific case and call it a "monotone sequence". Definition: it is fair to call any sequence monotonically increasing if it satisfies the strict inequality x n< 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 is easier to understand this with examples. The sequence given by the formula x n \u003d 2 + n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence. And if we take x n \u003d 1 / n, then we get a series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing 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 quantity (real or complex). If you draw a sequence diagram, then at a certain point it will, as it were, converge, tend to turn into a certain value. Hence the name - convergent sequence. Such a sequence may or may not have a limit. First, it is useful to understand when it is, from here you can start when proving the absence of a limit. Among monotonic sequences, convergent and divergent are distinguished. Convergent - this is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent - a sequence that has no limit in its set (neither real nor complex). Moreover, the sequence converges if its upper and lower limits converge in a geometric representation. The limit of a convergent sequence can in many cases be equal to zero, since any infinitesimal sequence has a known limit (zero). Whichever convergent sequence you take, they are all bounded, but far from all bounded sequences converge. The sum, difference, product of two convergent sequences is also a convergent sequence. However, the quotient can also converge if it is defined! Limits of sequences are the same significant (in most cases) value as numbers and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits. First, just 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 is true: 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 equal to zero. After all, if the limit of sequences is equal to zero, then division by zero will turn out, which is impossible. It would seem that the limit of the numerical sequence has already been analyzed in some detail, but such phrases 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 infinitely small, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such values have their own characteristics. The properties of the limit of a sequence having arbitrary small or large values are as follows: In fact, calculating the limit of a sequence is not such a difficult task if you know a simple algorithm. But the limits of sequences are a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution of such expressions. Starting small, over time, you can reach big heights. Definition. If each natural number n is assigned a number xn, then we say that a sequence is given x1, x2, …, xn = (xn) The common element of the sequence is a function of n. Thus a sequence can be viewed as a function. You can specify a sequence in various ways - the main thing is that a method for obtaining any member of the sequence is indicated. Example. (xn) = ((-1)n) or (xn) = -1; one; -one; one; … (xn) = (sinn/2) or (xn) = 1; 0; one; 0; … You can define the following operations for sequences: Multiplication of a sequence by a number m: m(xn) = (mxn), i.e. mx1, mx2, … Addition (subtraction) of sequences: (xn) (yn) = (xn yn). Product of sequences: (xn)(yn) = (xnyn). Quotient of sequences: at (yn) 0. Bounded and unbounded sequences. Definition. A sequence (xn) is called bounded if there exists a number M>0 such that for any n the inequality is true: those. all members of the sequence belong to the interval (-M; M). Definition. A sequence (xn) is said to be bounded from above if for any n there exists a number M such that xn M. Definition. A sequence (xn) is said to be bounded from below if for any n there exists a number M such that xn M Example. (xn) = n - bounded from below (1, 2, 3, … ). Definition. The number a is called the limit of the sequence (xn) if for any positive >0 there is such a number N that for all n > N the condition is satisfied: This is written: lim xn = a. In this case, the sequence (xn) is said to converge to a for n. Property: If we discard any number of members of the sequence, then new sequences are obtained, and if one of them converges, then the other also converges. Example. Prove that the limit of the sequence is lim . Let it be true for n > N, i.e. . This is true for , so if N is taken as the integer part of , then the above statement is true. Example. Show that for n the sequence is 3, has a limit of 2. Total: (xn)= 2 + 1/n; 1/n = xn - 2 Obviously, there exists a number n such that, i.e. lim (xn) = 2. Theorem. A sequence cannot have more than one limit. Proof. Assume that the sequence (xn) has two limits a and b that are not equal to each other. xn a; xnb; a b. Then by definition there exists a number >0 such that Definition. The sequence (x n ) is called limited, if there is such a number M>0 that for any n the following inequality is true: those. all members of the sequence belong to the interval (-M; M). Definition. The sequence (x n) is called bounded from above Definition. The sequence (x n ) is called bounded from below if for any n there exists a number M such that Example. (x n ) = n - bounded from below (1, 2, 3,). Definition. Number a is called the limit of the sequence (x n ) if for any positive e>0 there is such a number N that for all n> N the condition is satisfied: This is written: lim x n = a. In this case, the sequence (x n ) is said to be converges to a for n®¥. Property: If we discard any number of members of the sequence, then new sequences are obtained, and if one of them converges, then the other also converges. Example. Prove that the limit of the sequence is lim . Let be true for n > N, i.e. . This is true for , so if N is the integer part of , then the statement above is true. Example. Show that for n®¥ the sequence 3 has the number 2 as its limit. Total: (x n)= 2 + 1/n; 1/n = x n - 2 Obviously, there exists a number n such that , i.e. lim(xn) = 2. Theorem. A sequence cannot have more than one limit. Proof. Suppose that the sequence (x n ) has two limits a and b that are not equal to each other. x n ® a; x n ® b; a ¹ b. Then, by definition, there exists a number e >0 such that Let's write the expression: And since e- any number, then , i.e. a = b. The theorem has been proven. Theorem. If x n ® a, then . Proof. From x n ® a follows that . In the same time: Those. , i.e. . The theorem has been proven. Theorem. If x n ® a, then the sequence (x n ) is bounded. It should be noted that the converse statement is not true, i.e. the boundedness of a sequence does not imply its convergence. For example, the sequence has no limit, although Definition: 1) If x n +1 > x n for all n, then the sequence is increasing. 2) If x n +1 ³ x n for all n, then the sequence is non-decreasing. 3) If x n +1< x n для всех n, то последовательность убывающая. 4) If x n +1 £ x n for all n, then the sequence is non-increasing All these sequences are called monotonous. Increasing and decreasing sequences are called strictly monotonous. Example. (x n ) = 1/n - decreasing and limited (x n ) = n - increasing and unlimited. Example. Prove that the sequence (x n ) = monotonic increasing. Let's find a member of the sequence (x n +1) = Find the sign of the difference: (x n)-(x n +1) = Because nнN, then the denominator is positive for any n. Thus, x n +1 > x n . The sequence is increasing, which should be proved. Example. Find out whether the sequence (x n ) = is increasing or decreasing. Let's find . Let's find the difference Because nнN, then 1 - 4n<0, т.е. х n +1 < x n . Последовательность монотонно убывает. It should be noted that monotonic sequences are limited on at least one side. Theorem. A monotone bounded sequence has a limit. Proof. Consider a monotone non-decreasing sequence x 1 £ x 2 £ x 3 £ … £ x n £ x n +1 £ This sequence is bounded from above: x n £ M, where M is some number. Because any numerical set bounded from above has a clear upper bound, then for any e>0 there exists a number N such that x N > a - e, where a is some upper bound of the set. Because (x n ) is a non-decreasing sequence, then for N > n a - e From here a - e< x n < a + e E< x n - a < e или ôx n - aô< e, т.е. lim x n = a. For other monotone sequences, the proof is similar. Introduction…………………………………………………………………………………3 1.Theoretical part………………………………………………………………….4 Basic concepts and terms…………………………………………………....4 1.1 Types of sequences……………………………………………………...6 1.1.1.Limited and unlimited number sequences…..6 1.1.2.Monotonicity of sequences……………………………………6 1.1.3.Infinitesimal and infinitesimal sequences…….7 1.1.4. Properties of infinitesimal sequences…………………8 1.1.5 Convergent and divergent sequences and their properties..…9 1.2 Sequence Limit…………………………………………………….11 1.2.1.Theorems about the limits of sequences………………………………………………………………15 1.3.Arithmetic progression…………………………………………………………17 1.3.1. Properties of an arithmetic progression……………………………………..17 1.4Geometric progression……………………………………………………..19 1.4.1. Properties of a geometric progression……………………………………….19 1.5. Fibonacci numbers………………………………………………………………..21 1.5.1 Connection of Fibonacci numbers with other areas of knowledge…………………….22 1.5.2. Using a series of Fibonacci numbers to describe animate and inanimate nature……………………………………………………………………………….23 2. Own research…………………………………………………….28 Conclusion………………………………………………………………………….30 List of used literature…………………………………………....31 Introduction. Number sequences are a very interesting and informative topic. This topic is found in tasks of increased complexity, which are offered to students by the authors of didactic materials, in the tasks of mathematical Olympiads, entrance exams to higher educational institutions and the USE. I am interested to know the connection of mathematical sequences with other fields of knowledge. The purpose of the research work: To expand knowledge about the numerical sequence. 1. Consider the sequence; 2. Consider its properties; 3. Consider the analytical task of the sequence; 4. Demonstrate its role in the development of other areas of knowledge. 5. Demonstrate the use of a series of Fibonacci numbers to describe animate and inanimate nature. 1. Theoretical part. Basic concepts and terms. Definition. A numerical sequence is a function of the form y = f(x), x О N, where N is the set of natural numbers (or a function of a natural argument), denoted y = f(n) or y1, y2,…, yn,…. The values y1, y2, y3,… are called respectively the first, second, third, … members of the sequence. The number a is called the limit of the sequence x = (x n ) if for an arbitrary pre-specified arbitrarily small positive number ε there is such natural number N such that for all n>N the inequality |x n - a|< ε. If the number a is the limit of the sequence x \u003d (x n), then they say that x n tends to a, and write A sequence (yn) is called increasing if each of its members (except the first) is greater than the previous one: y1< y2 < y3 < … < yn < yn+1 < …. A sequence (yn) is called decreasing if each of its members (except the first) is less than the previous one: y1 > y2 > y3 > … > yn > yn+1 > … . Increasing and decreasing sequences are united by a common term - monotonic sequences. A sequence is called periodic if there exists a natural number T such that, starting from some n, the equality yn = yn+T holds. The number T is called the period length. An arithmetic progression is a sequence (an), each member of which, starting from the second, is equal to the sum of the previous member and the same number d, is called an arithmetic progression, and the number d is called the difference of an arithmetic progression. In this way, arithmetic progression- this is numerical sequence(an) given recursively by the relations a1 = a, an = an–1 + d (n = 2, 3, 4, …) A geometric progression is a sequence in which all members are non-zero and each member of which, starting from the second, is obtained from the previous member by multiplying by the same number q. Thus, a geometric progression is a numerical sequence (bn) given recursively by the relations b1 = b, bn = bn–1 q (n = 2, 3, 4…). 1.1 Types of sequences. 1.1.1 Bounded and unbounded sequences. A sequence (bn) is said to be bounded from above if there exists a number M such that for any number n the inequality bn≤ M is satisfied; A sequence (bn) is said to be bounded from below if there exists a number M such that for any number n the inequality bn≥ M is satisfied; For example: 1.1.2 Monotonicity of sequences. A sequence (bn) is called nonincreasing (nondecreasing) if for any number n the inequality bn≥ bn+1 (bn ≤bn+1) is true; A sequence (bn) is called decreasing (increasing) if for any number n the inequality bn > bn+1 (bn Decreasing and increasing sequences are called strictly monotonic, non-increasing - monotonic in a broad sense. Sequences bounded both above and below are called bounded. The sequence of all these types is called monotonic. 1.1.3 Infinitely large and small sequences. An infinitesimal sequence is a numerical function or sequence that tends to zero. A sequence an is called infinitesimal if A function is called infinitesimal in a neighborhood of the point x0 if ℓimx→x0 f(x)=0. A function is called infinitesimal at infinity if ℓimx→.+∞ f(x)=0 or ℓimx→-∞ f(x)=0 Also infinitesimal is a function that is the difference between a function and its limit, that is, if ℓimx→.+∞ f(x)=а, then f(x) − a = α(x), ℓimx→.+∞ f(( x)-a)=0. An infinitely large sequence is a numerical function or sequence that tends to infinity. A sequence an is called infinitely large if ℓimn→0 an=∞. A function is called infinite in a neighborhood of a point x0 if ℓimx→x0 f(x)= ∞. A function is said to be infinitely large at infinity if ℓimx→.+∞ f(x)= ∞ or ℓimx→-∞ f(x)= ∞ . 1.1.4 Properties of infinitesimal sequences. The sum of two infinitesimal sequences is itself also an infinitesimal sequence. The difference of two infinitesimal sequences is itself also an infinitesimal sequence. The algebraic sum of any finite number of infinitesimal sequences is itself also an infinitesimal sequence. The product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence. The product of any finite number of infinitesimal sequences is an infinitesimal sequence. Any infinitesimal sequence is bounded. If the stationary sequence is infinitely small, then all its elements, starting from some, are equal to zero. If the entire infinitesimal sequence consists of the same elements, then these elements are zeros. If (xn) is an infinitely large sequence containing no zero terms, then there is a sequence (1/xn) that is infinitesimal. If, however, (xn) contains zero elements, then the sequence (1/xn) can still be defined starting from some number n, and will still be infinitesimal. If (an) is an infinitesimal sequence containing no zero terms, then there is a sequence (1/an) that is infinitely large. If, however, (an) contains zero elements, then the sequence (1/an) can still be defined starting from some number n, and will still be infinitely large. 1.1.5 Convergent and divergent sequences and their properties. A convergent sequence is a sequence of elements of the set X that has a limit in this set. A divergent sequence is a sequence that is not convergent. Every infinitesimal sequence is convergent. Its limit is zero. Removing any finite number of elements from an infinite sequence does not affect either the convergence or the limit of that sequence. Any convergent sequence is bounded. However, not every bounded sequence converges. If the sequence (xn) converges, but is not infinitely small, then, starting from some number, the sequence (1/xn) is defined, which is bounded. The sum of convergent sequences is also a convergent sequence. The difference of convergent sequences is also a convergent sequence. The product of convergent sequences is also a convergent sequence. The quotient of two convergent sequences is defined starting from some element, unless the second sequence is infinitesimal. If the quotient of two convergent sequences is defined, then it is a convergent sequence. If a convergent sequence is bounded below, then none of its lower bounds exceeds its limit. If a convergent sequence is bounded from above, then its limit does not exceed any of its upper bounds. If for any number the terms of one convergent sequence do not exceed the terms of another convergent sequence, then the limit of the first sequence also does not exceed the limit of the second.Determining the Limit of a Sequence
General notation for the limit of sequences
Uncertainty and certainty of the limit
What is a neighborhood?
Theorems
Sequence Proof
Or maybe he doesn't exist?
monotonic sequence
Limit of convergent and bounded sequence
Monotonic sequence limit
Various actions with limits
Sequence Value Properties
Monotonic sequences