Number sequence and its limit represent one of the most important problems of mathematics throughout the history of the existence of this science. Constantly updated knowledge, formulated new theorems and proofs - all this allows us to consider this concept from new positions and under different
A number sequence, according to one of the most common definitions, is a mathematical function, the basis of which is a set of natural numbers arranged according to one or another pattern.
There are several options for creating number sequences.
Firstly, this function can be specified in the so-called “explicit” way, when there is a certain formula with the help of which each of its members can be determined by simply substituting a serial number into a given sequence.
The second method is called “recurrent”. Its essence is that the first few terms of a numerical sequence are specified, as well as a special recurrent formula, with the help of which, knowing the previous term, you can find the next one.
Finally, most in a general way assignment of sequences is the so-called when, without much difficulty, you can not only identify one or another member under a certain serial number, but also, knowing several consecutive members, come to a general formula for a given function.
The number sequence can be decreasing or increasing. In the first case, each subsequent member is less than the previous one, and in the second, on the contrary, it is greater.
Considering this topic, we cannot help but touch upon the question of the limits of sequences. The limit of a sequence is a number when, for any value, including an infinitesimal one, there is an ordinal number after which the deviation of successive members of the sequence from a given point in numerical form becomes less than the value specified during the formation of this function.
The concept of the limit of a numerical sequence is actively used when carrying out certain integral and differential calculations.
Mathematical sequences have a whole set of quite interesting properties.
Firstly, any number sequence is an example of a mathematical function, therefore, those properties that are characteristic of functions can be safely applied to sequences. The most a shining example such properties is the provision on increasing and decreasing arithmetic series, which are united by one general concept- monotonic sequences.
Secondly, there is enough large group sequences that cannot be classified as either increasing or decreasing are periodic sequences. In mathematics, they are usually considered to be those functions in which the so-called period length exists, that is, from a certain moment (n) the following equality y n = y n+T begins to apply, where T will be the very length of the period.
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 to add, subtract, multiply and divide, to high school come into play letter designations, and in the older age you can’t do without them.
But today we will talk about what all known mathematics is based on. About a community of numbers called “sequence limits”.
What are sequences and where is their limit?
The meaning of the word “sequence” is not difficult to interpret. This is an arrangement of things where someone or something is located in a certain order or queue. For example, the queue for tickets to the zoo is a sequence. And there can only be one! If, for example, you look at the queue at the store, this is one sequence. And if one person from this queue suddenly leaves, then this is a different queue, a different order.
The word “limit” is also easily interpreted - it is the end of something. However, in mathematics, the limits of sequences are those values on the number line to which a sequence of numbers tends. Why does it strive and not end? It's simple, the number line has no end, and most sequences, like rays, have only a beginning and look like this:
x 1, x 2, x 3,...x n...
Hence the definition of a sequence is a function of the natural argument. More in simple words is a series of members of a certain set.
How is the number sequence constructed?
A simple example of a number sequence might look like this: 1, 2, 3, 4, …n…
In most cases, for practical purposes, sequences are built from numbers, and each next member of the series, let's denote it by X, has its own name. For example:
x 1 is the first member of the sequence;
x 2 is the second term of the sequence;
x 3 is the third term;
x n is the nth term.
In practical methods, the sequence is given by a general formula in which there is a certain variable. For example:
X n =3n, then the series of numbers itself will look like this:
It is worth remembering that when writing sequences in general, you can use any Latin letters, not just X. For example: y, z, k, etc.
Arithmetic progression as part of sequences
Before looking for the limits of sequences, it is advisable to plunge deeper into the very concept of such a number series, which everyone encountered when they were in middle school. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.
Problem: “Let a 1 = 15, and the progression step of the number series d = 4. Construct the first 4 terms of this series"
Solution: a 1 = 15 (by condition) is the first term of the progression (number series).
and 2 = 15+4=19 is the second term of the progression.
and 3 =19+4=23 is the third term.
and 4 =23+4=27 is the fourth term.
However similar method it is difficult to reach large values, for example up to a 125. . Especially for such cases, a formula convenient for practice was derived: a n =a 1 +d(n-1). In this case, a 125 =15+4(125-1)=511.
Types of sequences
Most of the sequences are endless, it's worth remembering for the rest of your life. There are two interesting looking number series. The first is given by the formula a n =(-1) n. Mathematicians often call this sequence a flasher. Why? Let's check its number series.
1, 1, -1, 1, -1, 1, etc. With an example like this, it becomes clear that numbers in sequences can easily be repeated.
Factorial sequence. It's easy to guess - the formula defining the sequence contains a factorial. For example: a n = (n+1)!
Then the sequence will look like this:
a 2 = 1x2x3 = 6;
and 3 = 1x2x3x4 = 24, etc.
A sequence defined by an arithmetic progression is called infinitely decreasing if the inequality -1 is satisfied for all its terms and 3 = - 1/8, etc. There is even a sequence consisting of the same number. So, n =6 consists of an infinite number of sixes. Sequence limits have long existed in mathematics. Of course, they deserve their own competent design. So, time to learn the definition of sequence limits. First, let's look at the limit for a linear function in detail: 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. A simple example: a x = 4x+1. Then the sequence itself will look like this. 5, 9, 13, 17, 21…x… Thus, this sequence will increase indefinitely, which means its limit is equal to infinity as x→∞, and it should be written like this: If we take a similar sequence, but x tends to 1, we get: And the series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number closer to one (0.1, 0.2, 0.9, 0.986). From this series it is clear that the limit of the function is five. From this part it is worth remembering what the limit of a numerical sequence is, the definition and method for solving simple problems. Having examined the limit of a number sequence, its definition and examples, you can proceed to a more complex topic. Absolutely all limits of sequences can be formulated by one formula, which is usually analyzed in the first semester. So, what does this set of letters, modules and inequality signs mean? ∀ is a universal quantifier, replacing the phrases “for all”, “for everything”, etc. ∃ is an existential quantifier, in this case it means that there is some value N belonging to the set of natural numbers. A long vertical stick following N means that the given set N is “such that.” In practice, it can mean “such that”, “such that”, etc. To reinforce the material, read the formula out loud. The method of finding the limit of sequences, which was discussed above, although simple to use, is not so rational in practice. Try to find the limit for this function: If we substitute different values of “x” (increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. This results in a rather strange fraction: But is this really so? Calculating the limit of a number sequence in this case seems quite easy. It would be possible to leave everything as it is, because the answer is ready, and it was received under reasonable conditions, but there is another way specifically for such cases. First, let's find the highest degree in the numerator of the fraction - this is 1, since x can be represented as x 1. Now let's find the highest degree in the denominator. Also 1. Let's divide both the numerator and the denominator by the variable to the highest degree. In this case, divide the fraction by x 1. Next, we will find what value each term containing a variable tends to. In this case, fractions are considered. As x→∞, the value of each fraction tends to zero. When submitting your work in writing, you should make the following footnotes: This results in the following expression: Of course, the fractions containing x did not become zeros! But their value is so small that it is completely permissible not to take it into account in calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero. Suppose the professor has at his disposal a complex sequence, given, obviously, by an equally complex formula. The professor has found the answer, but is it right? After all, all people make mistakes. Auguste Cauchy once came up with an excellent way to prove the limits of sequences. His method was called neighborhood manipulation. Suppose that there is a certain point a, its neighborhood in both directions on the number line is equal to ε (“epsilon”). Since the last variable is distance, its value is always positive. Now let's define some sequence x n and assume that the tenth term of the sequence (x 10) is in the neighborhood of a. How can we write this fact in mathematical language? Let's say x 10 is to the right of point a, then the distance x 10 -a<ε, однако, если расположить «икс десятое» левее точки а, то расстояние получится отрицательным, а это невозможно, значит, следует занести левую часть неравенства под модуль. Получится |х 10 -а|<ε. Now it’s time to explain in practice the formula discussed above. It is fair to call a certain number a the end point of a sequence if for any of its limits the inequality ε>0 is satisfied, and the entire neighborhood has its own natural number N, such that all members of the sequence with higher numbers will be inside the sequence |x n - a|< ε. With such knowledge it is easy to solve the sequence limits, prove or disprove the ready-made answer. Theorems on the limits of sequences are an important component of the theory, without which practice is impossible. There are only four main theorems, remembering which can make the process of solving or proving much easier: Sometimes you need to solve an inverse problem, to prove a given limit of a numerical sequence. Let's look at an example. Prove that the limit of the sequence given by the formula is zero. According to the rule discussed above, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим: Let us express n through “epsilon” to show the existence of a certain number and prove the presence of a limit of the sequence. At this point, it is important to remember that “epsilon” and “en” are positive numbers and are not equal to zero. Now it is possible to continue further transformations using the knowledge about inequalities gained in high school. How does it turn out that n > -3 + 1/ε. Since it is worth remembering that we are talking about natural numbers, the result can be rounded by putting it in square brackets. Thus, it was proven that for any value of the “epsilon” neighborhood of the point a = 0, a value was found such that the initial inequality is satisfied. From here we can safely say that the number a is the limit of a given sequence. Q.E.D. This convenient method can be used to prove the limit of a numerical sequence, no matter how complex it may be at first glance. The main thing is not to panic when you see the task. The existence of a consistency limit is not necessary in practice. You can easily come across series of numbers that really have no end. For example, the same “flashing light” x n = (-1) n. it is obvious that a sequence consisting of only two digits, repeated cyclically, cannot have a limit. The same story is repeated with sequences consisting of one number, fractional ones, having uncertainty of any order during calculations (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculations also occur. Sometimes double-checking your own solution will help you find the sequence limit. Several examples of sequences and methods for solving them were discussed above, and now let’s try to take a more specific case and call it a “monotonic sequence.” Definition: any sequence can rightly be called monotonically increasing if the strict inequality x n holds for it< x n +1. Также любую последовательность справедливо называть монотонной убывающей, если для неё выполняется неравенство x n >x n +1. Along with these two conditions, there are also similar non-strict inequalities. Accordingly, x n ≤ x n +1 (non-decreasing sequence) and x n ≥ x n +1 (non-increasing sequence). But it’s easier to understand this with examples. The sequence given by the formula x n = 2+n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence. And if we take x n =1/n, we get the series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing sequence. A bounded sequence is a sequence that has a limit. A convergent sequence is a series of numbers that has an infinitesimal limit. Thus, the limit of a bounded sequence is any real or complex number. Remember that there can only be one limit. The limit of a convergent sequence is an infinitesimal (real or complex) quantity. If you draw a sequence diagram, then at a certain point it will seem to converge, tend to turn into a certain value. Hence the name - convergent sequence. There may or may not be a limit to such a sequence. First, it is useful to understand when it exists; from here you can start when proving the absence of a limit. Among monotonic sequences, convergent and divergent are distinguished. Convergent is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent is a sequence that has no limit in its set (neither real nor complex). Moreover, the sequence converges if, in a geometric representation, its upper and lower limits converge. The limit of a convergent sequence can be zero in many cases, since any infinitesimal sequence has a known limit (zero). Whatever convergent sequence you take, they are all bounded, but not all bounded sequences converge. The sum, difference, product of two convergent sequences is also a convergent sequence. However, the quotient can also be convergent if it is defined! Sequence limits are as significant (in most cases) as digits and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits. First, like digits and numbers, the limits of any sequence can be added and subtracted. Based on the third theorem on the limits of sequences, the following equality holds: the limit of the sum of sequences is equal to the sum of their limits. Secondly, based on the fourth theorem on the limits of sequences, the following equality is true: the limit of the product of the nth number of sequences equal to the product their limits. The same is true for division: the limit of the quotient of two sequences is equal to the quotient of their limits, provided that the limit is not zero. After all, if the limit of sequences is equal to zero, then division by zero will result, which is impossible. It would seem that the limit of the numerical sequence has already been discussed in some detail, but phrases such as “infinitely small” and “infinitely large” numbers are mentioned more than once. Obviously, if there is a sequence 1/x, where x→∞, then such a fraction is infinitesimal, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such quantities have their own characteristics. The properties of the limit of a sequence having any small or large values are as follows: In fact, calculating the limit of a sequence is not such a difficult task if you know a simple algorithm. But the limits of consistency are a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution to such expressions. Starting small, you can achieve great heights over time. Cradle. Diapers. Cry. Life sequence SEQUENCE - numbers or elements arranged in an organized order. 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 Definition:Numerical sequence is called a numeric given on the set N of natural numbers. For numerical sequences, usually instead of f(n) write a n
and denote the sequence as follows: ( a n
). Numbers a 1
,
a 2
, …,
a n,…
called elements of the sequence. Usually the number sequence is determined by the task n th element or a recurrent formula by which each subsequent element is determined through the previous one. A descriptive way of specifying a numerical sequence is also possible. For example: With the recurrent method, indicate a formula that allows you to express n th member of the sequence through the previous ones, and specify 1–2 initial members of the sequence. Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; …. Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8; Sequence expressed by recurrence formula y n =y n-1
+ 4
can also be specified analytically: y n= y 1 +4*(n-1)
Let's check: y2=3+4*(2-1)=7, y3=3+4*(3-1)=11 Here we do not need to know the previous member of the numerical sequence to calculate the nth element; we just need to specify its number and the value of the first element. As we can see, this method of specifying a numerical sequence is very similar to the analytical method of specifying functions. In fact, a number sequence is a special type of number function, so a number of properties of functions can be considered for sequences as well. 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. And if you want to learn more about the different types of number sequences, click here. Well, if everything is clear and simple to you, then try to answer. 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 number series 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 such natural number N 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, arithmetic progression is a numerical sequence (an) defined recursively 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 some, 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. Vida y= f(x), x ABOUT N,
Where N– a set of natural numbers (or a function of a natural argument), denoted y=f(n)
or y 1 ,y 2 ,…, y n,…. Values y 1 ,y 2 ,y 3 ,…
are called respectively the first, second, third, ... members of the sequence. For example, for the function y= n 2 can be written: y 1 = 1 2 = 1; y 2 = 2 2 = 4; y 3 = 3 2 = 9;…y n = n 2 ;… Methods for specifying sequences. Sequences can be specified in various ways, among which three are especially important: analytical, descriptive and recurrent.
1. A sequence is given analytically if its formula is given n th member: y n=f(n). Example. y n= 2n – 1 –
sequence of odd numbers: 1, 3, 5, 7, 9, … 2. Descriptive
The way to specify a numerical sequence is to explain from which elements the sequence is built. Example 1. “All terms of the sequence are equal to 1.” This means we are talking about a stationary sequence 1, 1, 1, …, 1, …. Example 2: “The sequence consists of all prime numbers in ascending order.” Thus, the given sequence is 2, 3, 5, 7, 11, …. With this method of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to. 3. The recurrent method of specifying a sequence is to specify a rule that allows you to calculate n-th member of a sequence if its previous members are known. The name recurrent method comes from the Latin word recurrent- come back. Most often in such cases a formula is indicated that allows one to express n th member of the sequence through the previous ones, and specify 1–2 initial members of the sequence. Example 1. y 1 = 3; y n = y n–1 + 4 if n = 2, 3, 4,…. Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; …. You can see that the sequence obtained in this example can also be specified analytically: y n= 4n – 1. Example 2. y 1 = 1; y 2 = 1; y n = y n –2 + y n–1 if n = 3, 4,…. Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8; The sequence in this example is especially studied in mathematics because it has a number of interesting properties and applications. It is called the Fibonacci sequence, named after the 13th century Italian mathematician. It is very easy to define the Fibonacci sequence recurrently, but very difficult analytically. n The th Fibonacci number is expressed through its serial number by the following formula. At first glance, the formula for n th Fibonacci number seems implausible, since the formula that specifies the sequence of natural numbers only contains square roots, but you can check “manually” the validity of this formula for the first few n. A numerical sequence is a special case of a numerical function, therefore a number of properties of functions are also considered for sequences. Definition .
Subsequence ( y n}
is called increasing if each of its terms (except the first) is greater than the previous one: y 1 y 2 y 3 y n y n +1 Definition.Sequence ( y n}
is called decreasing if each of its terms (except the first) is less than the previous one: y 1 > y 2 > y 3 > … > y n> y n +1 > …
. Increasing and decreasing sequences are combined under the common term - monotonic sequences. Example 1. y 1 = 1; y n= n 2 – increasing sequence. Thus, the following theorem is true (a characteristic property of an arithmetic progression). A number sequence is arithmetic if and only if each of its members, except the first (and the last in the case of a finite sequence), is equal to the arithmetic mean of the preceding and subsequent members. Example. At what value x numbers 3 x + 2, 5x– 4 and 11 x+ 12 form a finite arithmetic progression? According to the characteristic property, the given expressions must satisfy the relation 5x – 4 = ((3x + 2) + (11x + 12))/2. Solving this equation gives x= –5,5.
At this value x given expressions 3 x + 2, 5x– 4 and 11 x+ 12 take, respectively, the values –14.5,
–31,5, –48,5.
This is an arithmetic progression, its difference is –17. A numerical sequence, all terms of which are non-zero and each term of which, starting from the second, is obtained from the previous term by multiplying by the same number q, is called a geometric progression, and the number q- the denominator of a geometric progression. Thus, a geometric progression is a number sequence ( b n), defined recursively by the relations b 1 = b, b n = b n –1 q (n = 2, 3, 4…). (b And q – given numbers, b ≠ 0, q ≠ 0). Example 1. 2, 6, 18, 54, ... – increasing geometric progression b = 2, q = 3. Example 2. 2, –2, 2, –2, … –
geometric progression b= 2,q= –1. Example 3. 8, 8, 8, 8, … –
geometric progression b= 8, q= 1. A geometric progression is an increasing sequence if b 1 > 0, q> 1, and decreasing if b 1 > 0, 0 q One of the obvious properties of a geometric progression is that if the sequence is a geometric progression, then so is the sequence of squares, i.e. b 1 2 , b 2 2 , b 3 2 , …, b n 2,... is a geometric progression whose first term is equal to b 1 2 , and the denominator is q 2 . Formula n- the th term of the geometric progression has the form b n= b 1 qn– 1 . You can obtain a formula for the sum of terms of a finite geometric progression. Let a finite geometric progression be given b 1 ,b 2 ,b 3 , …, b n let S n – the sum of its members, i.e. S n= b 1 + b 2 + b 3 + … +b n. It is accepted that q No. 1. To determine S n an artificial technique is used: some geometric transformations of the expression are performed S n q. S n q = (b 1 + b 2 + b 3 + … + b n –1 + b n)q = b 2 + b 3 + b 4 + …+ b n+ b n q = S n+ b n q– b 1 . Thus, S n q= S n +b n q – b 1 and therefore This is the formula with umma n terms of geometric progression for the case when q≠ 1. At q= 1 the formula need not be derived separately; it is obvious that in this case S n= a 1 n. The progression is called geometric because each term in it, except the first, is equal to the geometric mean of the previous and subsequent terms. Indeed, since bn=bn- 1 q; bn = bn+ 1 /q, hence, b n 2=bn– 1 bn+ 1 and the following theorem is true (a characteristic property of a geometric progression): a number sequence is a geometric progression if and only if the square of each of its terms, except the first (and the last in the case of a finite sequence), is equal to the product of the previous and subsequent terms. Let there be a sequence ( c n} = {1/n}.
This sequence is called harmonic, since each of its terms, starting from the second, is the harmonic mean between the previous and subsequent terms. Geometric mean of numbers a And b there is a number Otherwise the sequence is called divergent. Based on this definition, one can, for example, prove the existence of a limit A=0 for the harmonic sequence ( c n} =
{1/n). Let ε be arbitrarily small positive number. The difference is considered Does such a thing exist? N that's for everyone n ≥ N inequality 1 holds /N ? If we take it as N any natural number greater than
1/ε
, then for everyone n ≥ N inequality 1 holds /n ≤
1/N ε,
Q.E.D.
Proving the presence of a limit for a particular sequence can sometimes be very difficult. The most frequently occurring sequences are well studied and are listed in reference books. There are important theorems that allow you to conclude that a given sequence has a limit (and even calculate it), based on already studied sequences. Theorem 1. If a sequence has a limit, then it is bounded. Theorem 2. If a sequence is monotonic and bounded, then it has a limit. Theorem 3. If the sequence ( a n}
has a limit A, then the sequences ( ca n}, {a n+ c)
and (| a n|}
have limits cA, A +c, |A| accordingly (here c– arbitrary number). Theorem 4. If the sequences ( a n}
And ( b n) have limits equal to A And B pa n + qbn) has a limit pA+ qB. Theorem 5. If the sequences ( a n) And ( b n)have limits equal to A And B accordingly, then the sequence ( a n b n) has a limit AB. Theorem 6. If the sequences ( a n}
And ( b n) have limits equal to A And B accordingly, and, in addition, b n ≠ 0 and B≠ 0, then the sequence ( a n / b n) has a limit A/B. Anna ChugainovaDetermining the Sequence Limit
General designation for the limit of sequences
Uncertainty and certainty of the limit
What is a neighborhood?
Theorems
Proof of sequences
Or maybe he's not there?
Monotonic sequence
Limit of a convergent and bounded sequence
Limit of a monotonic sequence
Various actions with limits
Properties of sequence quantities
Word. Step. Cold. Doctor.
Running around. Toys. Brother.
Yard Swing. Kindergarten.
School. Two. Troika. Five.
Ball. Step. Gypsum. Bed.
Fight. Blood. Broken nose.
Yard Friends. Party. Force.
Institute. Spring. Bushes.
Summer. Session. Tails.
Beer. Vodka. Gin with ice.
Coffee. Session. Diploma.
Romanticism. Love. Star.
Hands. Lips. A night without sleep.
Wedding. Mother-in-law. Father-in-law. Trap.
Argument. Club. Friends. Cup.
House. Job. House. Family.
Sun. Summer. Snow. Winter.
Son. Diapers. Cradle.
Stress. Mistress. Bed.
Business. Money. Plan. Emergency.
TV. Series.
Country house. Cherries. Zucchini.
Gray hair. Migraine. Glasses.
Grandson. Diapers. Cradle.
Stress. Pressure. Bed.
Heart. Kidneys. Bones. Doctor.
Speeches. Coffin. Farewell. Cry.
Properties of number sequences.
Geometric progression.
Consistency limit.
