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 is increasingly closer to 0 in location on the number line. 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. 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, ranging 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 coefficients of the Fibonacci series, 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. 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 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. 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. Subsequence Subsequence- This kit elements of some set: So the sequence turns out to be the result consistent selection of elements of a given set. And, if any set of elements is finite, and we talk about a sample of finite volume, then the sequence turns out to be a sample of infinite volume. A sequence is by its nature a mapping, so it should not be confused with a set that “runs through” the sequence. In mathematics, many different sequences are considered: The purpose of studying all possible sequences is to search for patterns, predict future states and generate sequences. Let a certain set of elements of arbitrary nature be given. | Any mapping from a set of natural numbers to a given set is called sequence(elements of the set). The image of a natural number, namely, the element, is called - th member or sequence element, and the ordinal number of a member of the sequence is its index. Sequences of the form It is customary to write compactly using parentheses: Curly braces are sometimes used: Allowing some freedom of speech, we can also consider finite sequences of the form which represent the image of the initial segment of a sequence of natural numbers. Wikimedia Foundation. 2010. SUBSEQUENCE. In I.V. Kireevsky’s article “The Nineteenth Century” (1830) we read: “From the very fall of the Roman Empire to our times, the enlightenment of Europe appears to us in gradual development and in uninterrupted sequence” (vol. 1, p.... ... History of words SEQUENCE, sequences, plural. no, female (book). distracted noun to sequential. A sequence of events. Consistency in the changing tides. Consistency in reasoning. Ushakov's Explanatory Dictionary.... ... Ushakov's Explanatory Dictionary Constancy, continuity, logic; row, progression, conclusion, series, string, turn, chain, chain, cascade, relay race; persistence, validity, set, methodicality, arrangement, harmony, tenacity, subsequence, connection, queue,... ... Synonym dictionary SEQUENCE, numbers or elements arranged in an organized manner. Sequences can be finite (having a limited number of elements) or infinite, such as the complete sequence of natural numbers 1, 2, 3, 4 ....... ... Scientific and technical encyclopedic dictionary SEQUENCE, a set of numbers (mathematical expressions, etc.; they say: elements of any nature), numbered by natural numbers. The sequence is written as x1, x2,..., xn,... or briefly (xi) ... Modern encyclopedia One of the basic concepts of mathematics. The sequence is formed by elements of any nature, numbered with natural numbers 1, 2, ..., n, ..., and written as x1, x2, ..., xn, ... or briefly (xn) ... Big Encyclopedic Dictionary Subsequence- SEQUENCE, a set of numbers (mathematical expressions, etc.; they say: elements of any nature), numbered with natural numbers. The sequence is written as x1, x2, ..., xn, ... or briefly (xi). ... Illustrated Encyclopedic Dictionary SEQUENCE, and, female. 1. See sequential. 2. In mathematics: an infinite ordered set of numbers. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary English succession/sequence; German Konsequenz. 1. The order of one after another. 2. One of the basic concepts of mathematics. 3. The quality of correct logical thinking, in which reasoning is free from internal contradictions in one and the other... ... Encyclopedia of Sociology Subsequence- “a function defined on the set of natural numbers, the set of values of which can consist of elements of any nature: numbers, points, functions, vectors, sets, random variables, etc., numbered by natural numbers... Economic-mathematical dictionary 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.Infinitely large 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 on the limits of sequences……………………………15 1.3. Arithmetic progression……………………………………………………………17 1.3.1. Properties of arithmetic progression…………………………………..17 1.4Geometric progression……………………………………………………………..19 1.4.1. Properties of 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 the Fibonacci number series to describe living and inanimate nature……………………………………………………………………………………………….23 2. Own research…………………………………………………….28 Conclusion………………………………………………………………………………….30 List of references……………………………………………………………....31 Introduction. Number sequences are a very interesting and educational topic. This topic is found in tasks of increased complexity that are offered to students by the authors of didactic materials, in problems of mathematical Olympiads, entrance exams to Higher Educational Institutions and the Unified State Exam. I'm interested in learning how mathematical sequences relate to other areas of knowledge. Purpose of the research work: To expand knowledge about the number 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 the Fibonacci series of numbers to describe living 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 a 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 the first, second, third,... members of the sequence, respectively. A number a is called the limit of the sequence x = (x n ) if for an arbitrary predetermined arbitrarily small positive number ε there is a 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 = (x n ), then they say that x n tends to a, and write A sequence (yn) is said to be increasing if each member (except the first) is greater than the previous one: y1< y2 < y3 < … < yn < yn+1 < …. A sequence (yn) is called decreasing if each member (except the first) is less than the previous one: y1 > y2 > y3 > … > yn > yn+1 > … . Increasing and decreasing sequences are combined under the common term - monotonic sequences. A sequence is called periodic if there is 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 term of which, starting from the second, is equal to the sum of the previous term and the same number d, is called an arithmetic progression, and the number d is the difference of an arithmetic progression. Thus, an arithmetic progression is a numerical sequence (an) defined recurrently by the relations a1 = a, an = an–1 + d (n = 2, 3, 4, …) A geometric progression is a sequence in which all terms are different from zero and each term of which, starting from the second, is obtained from the previous term by multiplying by the same number q. Thus, a geometric progression is a numerical sequence (bn) defined recurrently by the relations b1 = b, bn = bn–1 q (n = 2, 3, 4…). 1.1 Types of sequences. 1.1.1 Restricted and unrestricted sequences. A sequence (bn) is said to be bounded above if there is a number M such that for any number n the inequality bn≤ M holds; A sequence (bn) is called bounded below if there is a number M such that for any number n the inequality bn≥ M holds; For example: 1.1.2 Monotonicity of sequences. A sequence (bn) is called non-increasing (non-decreasing) 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 sequences are called monotonic in the broad sense. Sequences that are 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 said to be 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)=a, 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 said to be infinitely large if ℓimn→0 an=∞. A function is said to be infinitely large in a neighborhood of the 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 a stationary sequence is infinitesimal, then all its elements, starting from a certain point, are equal to zero. If the entire infinitesimal sequence consists of identical 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 (an) nevertheless 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 a 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 affects neither the convergence nor the limit of that sequence. Any convergent sequence is bounded. However, not every bounded sequence converges. If the sequence (xn) converges, but is not infinitesimal, then, starting from a certain number, a 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 at 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 infimums exceeds its limit. If a convergent sequence is bounded 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. Subsequence Subsequence noun, and., used compare often
Morphology: (no) what? sequences, what? sequences, (see) what? subsequence, how? sequence, about what? about sequence;
pl.
What? sequences, (no) what? sequences, what? sequences, (see) what? sequences, how? sequences, about what? about sequences
1.
Consistency called a row in which one element is located next to another. Continuous sequence. | Chronological sequence. | Remember the sequence of events. | Consistency in reasoning. | Consistency in actions.
2. In mathematics, computer science sequence name a series of numbers, information elements of a certain type. Infinite number sequence. | Consistency limit. | A structure is an object consisting of a sequence of named members, each member can be of any type.
Explanatory dictionary of the Russian language by Dmitriev. D. V. Dmitriev. 2003. A sequence is a set of elements of a certain set: for each natural number you can specify an element of this set; this number is the number of the element and indicates the position of this element in the sequence; for anyone... ... Wikipedia SUBSEQUENCE. In I.V. Kireevsky’s article “The Nineteenth Century” (1830) we read: “From the very fall of the Roman Empire to our times, the enlightenment of Europe appears to us in gradual development and in uninterrupted sequence” (vol. 1, p.... ... History of words SEQUENCE, sequences, plural. no, female (book). distracted noun to sequential. A sequence of events. Consistency in the changing tides. Consistency in reasoning. Ushakov's Explanatory Dictionary.... ... Ushakov's Explanatory Dictionary Constancy, continuity, logic; row, progression, conclusion, series, string, turn, chain, chain, cascade, relay race; persistence, validity, set, methodicality, arrangement, harmony, tenacity, subsequence, connection, queue,... ... Synonym dictionary SEQUENCE, numbers or elements arranged in an organized manner. Sequences can be finite (having a limited number of elements) or infinite, such as the complete sequence of natural numbers 1, 2, 3, 4 ....... ... Scientific and technical encyclopedic dictionary SEQUENCE, a set of numbers (mathematical expressions, etc.; they say: elements of any nature), numbered by natural numbers. The sequence is written as x1, x2,..., xn,... or briefly (xi) ... Modern encyclopedia One of the basic concepts of mathematics. The sequence is formed by elements of any nature, numbered with natural numbers 1, 2, ..., n, ..., and written as x1, x2, ..., xn, ... or briefly (xn) ... Big Encyclopedic Dictionary Subsequence- SEQUENCE, a set of numbers (mathematical expressions, etc.; they say: elements of any nature), numbered with natural numbers. The sequence is written as x1, x2, ..., xn, ... or briefly (xi). ... Illustrated Encyclopedic Dictionary SEQUENCE, and, female. 1. See sequential. 2. In mathematics: an infinite ordered set of numbers. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary English succession/sequence; German Konsequenz. 1. The order of one after another. 2. One of the basic concepts of mathematics. 3. The quality of correct logical thinking, in which reasoning is free from internal contradictions in one and the other... ... Encyclopedia of Sociology Subsequence- “a function defined on the set of natural numbers, the set of values of which can consist of elements of any nature: numbers, points, functions, vectors, sets, random variables, etc., numbered by natural numbers... Economic-mathematical dictionary Definition . The sequence is denoted as the nth term enclosed in curly braces: . The following designations are also possible: . They explicitly indicate that the index n belongs to the set of natural numbers and the sequence itself has an infinite number of terms. Here are some example sequences: In other words, a number sequence is a function whose domain of definition is the set of natural numbers. The number of elements of the sequence is infinite. Among the elements there may also be members that have the same meanings. Also, a sequence can be considered as a numbered set of numbers consisting of an infinite number of members. We will be mainly interested in the question of how sequences behave when n tends to infinity: . This material is presented in the section Limit of a sequence - basic theorems and properties. Here we will look at some examples of sequences. Consider the sequence. The common member of this sequence is . Let's write down the first few terms: Now consider a sequence with a common term. Here are its first few members: Consider the sequence. Her common member. The first terms have the following form: Next, consider the sequence. Her common member. Here are some of its first members: Consider a sequence with the following common term: Now let's look at a more interesting sequence. Let's take a segment on the number line. Let's divide it in half. We get two segments. Let As a result, we obtain a sequence whose elements are distributed in an open interval (0; 1)
. Whatever point we take from the closed interval
, we can always find members of the sequence that will be arbitrarily close to this point or coincide with it. Then from the original sequence one can select a subsequence that will converge to an arbitrary point from the interval
. That is, as the number n increases, the members of the subsequence will come closer and closer to the pre-selected point. For example, for point a = 0
you can choose the following subsequence: For point a = 1
Let's choose the following subsequence: Since there are subsequences that converge to different values, the original sequence itself does not converge to any number. Now let's construct a sequence that contains all rational numbers. Moreover, each rational number will appear in such a sequence an infinite number of times. The rational number r can be represented as follows: To do this, draw the p and q axes on the plane. We draw grid lines through the integer values of p and q. Then each node of this grid c will correspond to a rational number. The entire set of rational numbers will be represented by a set of nodes. We need to find a way to number all the nodes so that we don't miss any nodes. This is easy to do if you number the nodes by squares, the centers of which are located at the point (0; 0)
(see picture). In this case, the lower parts of the squares with q < 1
we don't need it. Therefore they are not shown in the figure. So, for the top side of the first square we have: In this way we obtain a sequence containing all rational numbers. You can notice that any rational number appears in this sequence an infinite number of times. Indeed, along with the node , this sequence will also include nodes , where is a natural number. But all these nodes correspond to the same rational number. Then from the sequence we have constructed, we can select a subsequence (having an infinite number of elements), all of whose elements are equal to a predetermined rational number. Since the sequence we constructed has subsequences that converge to different numbers, the sequence does not converge to any number. Here we have given a precise definition of the number sequence. We also raised the issue of its convergence, based on intuitive ideas. The exact definition of convergence is discussed on the page Defining the Limit of a Sequence. Related properties and theorems are stated on the page
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See what “Sequence” is in other dictionaries:
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See what “sequence” is in other dictionaries:
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Numerical sequence (xn)
is a law (rule) according to which, for every natural number n = 1, 2, 3, . . .
a certain number x n is assigned.
The element x n is called the nth member or element of the sequence.
,
,
.
Sequence Examples
Examples of infinitely increasing sequences
.
It can be seen that as the number n increases, the elements increase indefinitely towards positive values. We can say that this sequence tends to: for .
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As the number n increases, the elements of this sequence increase unlimitedly in absolute value, but do not have a constant sign. That is, this sequence tends to: at .Examples of sequences converging to a finite number
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It can be seen that as the number n increases, the elements of this sequence approach their limiting value a = 0
: at . So each subsequent term is closer to zero than the previous one. In a sense, we can consider that there is an approximate value for the number a = 0
with error. It is clear that as n increases, this error tends to zero, that is, by choosing n, the error can be made as small as desired. Moreover, for any given error ε > 0
you can specify a number N such that for all elements with numbers greater than N:, the deviation of the number from the limit value a will not exceed the error ε:.
.
In this sequence, terms with even numbers are equal to zero. Terms with odd n are equal. Therefore, as n increases, their values approach the limiting value a = 0
. This also follows from the fact that
.
Just like in the previous example, we can specify an arbitrarily small error ε > 0
, for which it is possible to find a number N such that elements with numbers greater than N will deviate from the limit value a = 0
by an amount not exceeding the specified error. Therefore this sequence converges to the value a = 0
: at .Examples of divergent sequences
Here are its first members:
.
It can be seen that terms with even numbers:
,
converge to the value a 1 = 0
. Odd-numbered members:
,
converge to the value a 2 = 2
. The sequence itself, as n grows, does not converge to any value.Sequence with terms distributed in the interval (0;1)
.
Let's divide each of the segments in half again. We get four segments. Let
.
Let's divide each segment in half again. Let's take
.
And so on.
.
= 0
.
.
The terms of this subsequence converge to the value a = 1
.
Sequence containing all rational numbers
,
where is an integer; - natural.
We need to associate each natural number n with a pair of numbers p and q so that any pair p and q is included in our sequence.
.
Next, we number the top part of the next square:
.
We number the top part of the following square:
.
And so on.Conclusion