Hovhannisyan Eva

Numeric sequences. Abstract.

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Municipal budgetary educational institution
"Average comprehensive school No. 31"
the city of Barnaul

Number Sequences

abstract

Work completed:
Oganesyan Eva,
8th grade student MBOU "Secondary School No. 31"
Supervisor:
Poleva Irina Alexandrovna,
mathematics teacher MBOU "Secondary School No. 31"

Barnaul - 2014

Introduction…………………………………………………………………………2

Numerical sequences.……………………………………………...3

Ways to set numerical sequences………………………...4

Development of the doctrine of progressions………………………………………………..5

Properties of numerical sequences……………………………………7

Arithmetic progression……………………………...................................................9

Geometric progression……………………………………………….10

Conclusion ………………………………………………………………… 11

References……………………………………………………………11

Introduction

Purpose of this abstract– study of the basic concepts related to numerical sequences, their application in practice.
Tasks:

1. To study the historical aspects of the development of the doctrine of progressions;
2. Consider ways of setting and properties of numerical sequences;
3. Learn about arithmetic and geometric progressions.

Currently, numerical sequences are considered as special cases of a function. The numerical sequence is a function of the natural argument. The concept of a numerical sequence arose and developed long before the creation of the theory of function. Here are examples of infinite number sequences known in antiquity:

1, 2, 3, 4, 5, … - sequence of natural numbers.

2, 4, 6, 8, 10,… - sequence even numbers.

1, 3, 5, 7, 9,… - a sequence of odd numbers.

1, 4, 9, 16, 25,… - sequence of squares of natural numbers.

2, 3, 5, 7, 11… - a sequence of prime numbers.

1, ½, 1/3, ¼, 1/5,… - sequence of reciprocals of natural numbers.

The number of members of each of these series is infinite; the first five sequences are monotonically increasing, the last one is monotonically decreasing. All of the listed sequences, except for the 5th one, are given due to the fact that for each of them the common term is known, i.e., the rule for obtaining a term with any number. For a sequence of prime numbers, a common term is unknown, but as early as the 3rd century. BC e. the Alexandrian scientist Eratosthenes indicated a method (albeit very cumbersome) for obtaining its n-th member. This method was called the "sieve of Eratosthenes".

Progressions - particular types of numerical sequences - are found in the monuments of the II millennium BC. e.

Number Sequences

There are various definitions of the number sequence.

Numeric sequence – it is a sequence of elements of a number space (Wikipedia).

Numeric sequence – this is a numbered number set.

A function of the form y = f (x), xis called a function of natural argument ornumerical sequenceand denote y = f(n) or

, , , …, The notation ().

We will write out positive even numbers in ascending order. The first such number is 2, the second is 4, the third is 6, the fourth is 8, and so on, so we get the sequence: 2; 4; 6; eight; 10 ….

Obviously, the fifth place in this sequence will be the number 10, the tenth - 20, the hundredth - 200. In general, for any natural number n, you can specify the corresponding positive even number; it is equal to 2n.

Let's look at another sequence. We will write out in descending order proper fractions with a numerator equal to 1:

; ; ; ; ; … .

For any natural number n, we can specify the corresponding fraction; it is equal to. So, in sixth place should be a fraction, on the thirtieth - , on the thousandth - a fraction .

The numbers that form a sequence are called the first, second, third, fourth, etc., respectively. members of the sequence. The members of a sequence are usually denoted by letters with subscripts indicating the ordinal number of the member. For instance:, , etc. in general, a member of the sequence with number n, or, as they say, nth member sequences denote. The sequence itself is denoted by (). A sequence can contain both an infinite number of members and a finite one. In this case, it is called final. For example: a sequence of two-digit numbers.10; eleven; 12; thirteen; …; 98; 99

Methods for specifying numerical sequences

Sequences can be specified in several ways.

Usually the sequence is more appropriate to setformula of its common nth term, which allows you to find any member of the sequence, knowing its number. In this case, the sequence is said to be given analytically. For example: a sequence of positive even terms=2n.

Task: find the formula for the common term of the sequence (:

6; 20; 56; 144; 352;…

Solution. We write each term of the sequence in the following form:

n=1: 6 = 2 3 = 3 =

n=2: 20=4 5=5=

n=3: 56 = 8 7 = 7 =

As you can see, the terms of the sequence are the product of a power of two multiplied by consecutive odd numbers, and two is raised to a power that is equal to the number of the element in question. Thus, we conclude that

Answer: common term formula:

Another way to specify a sequence is to specify a sequence usingrecurrent relation. A formula that expresses any member of a sequence, starting with some through the previous ones (one or more), is called recurrent (from the Latin word recurro - to return).

In this case, one or several first elements of the sequence are specified, and the rest are determined according to some rule.

An example of a recursively given sequence is the sequence of Fibonacci numbers - 1, 1, 2, 3, 5, 8, 13, ... , in which each subsequent number, starting from the third, is the sum of the two previous ones: 2 = 1 + 1; 3 = 2 + 1 and so on. This sequence can be given recursively:

N N, = 1.

Task: sequencegiven by the recurrence relation+ , n N, = 4. Write down the first few terms of this sequence.

Solution. Let's find the third term of the given sequence:

+ =

Etc.

When sequences are specified recurrently, calculations are very cumbersome, since in order to find elements with large numbers, it is necessary to find all previous members of the specified sequence, for example, to findwe need to find all the previous 499 terms.

Descriptive wayassignment of a numerical sequence consists in explaining what elements the sequence is built from.

Example 1 . "All members of the sequence are 1." This means that 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 sequence 2, 3, 5, 7, 11, … is given. 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.

Also, a numerical sequence can be given by a simplelisting its members.

Development of the doctrine of progressions

The word progression is of Latin origin (progressio), literally means “moving forward” (like the word “progress”) and is found for the first time by the Roman author Boethius (5th-6th centuries). continue it indefinitely in one direction, for example, a sequence of natural numbers, their squares and cubes. At the end of the Middle Ages and at the beginning of modern times, this term ceases to be commonly used. In the 17th century, for example, J. Gregory used the term "series" instead of progression, and another prominent English mathematician, J. Wallis, used the term "infinite progressions" for infinite series.

At present, we consider progressions as special cases of numerical sequences.

Theoretical information related to progressions is first found in the documents of ancient Greece that have come down to us.

In the Psammite, Archimedes for the first time compares arithmetic and geometric progressions:

1,2,3,4,5,………………..

10, , ………….

Progressions were considered as a continuation of proportions, which is why the epithets arithmetic and geometric were transferred from proportions to progressions.

This view of progressions was preserved by many mathematicians of the 17th and even 18th centuries. This is how one should explain the fact that the symbol found in Barrow, and then in other English scientists of that time to denote a continuous geometric proportion, began to denote a geometric progression in English and French textbooks of the 18th century. By analogy, they began to designate an arithmetic progression.

One of the proofs of Archimedes, set forth in his work "The Quadrature of the Parabola", essentially boils down to the summation of an infinitely decreasing geometric progression.

To solve some problems from geometry and mechanics, Archimedes derived the formula for the sum of the squares of natural numbers, although it was used before him.

1/6n(n+1)(2n+1)

Some formulas related to progressions were known to Chinese and Indian scientists. So, Aryabhatta (V century) knew the formulas for the common term, the sum of an arithmetic progression, etc., Magavira (IX century) used the formula: + + + ... + = 1/6n(n+1)(2n+1) and other more complex series. However, the rule for finding the sum of terms of an arbitrary arithmetic progression is first found in the Book of the Abacus (1202) by Leonardo of Pisa. In The Science of Numbers (1484), N. Shuke, like Archimedes, compares the arithmetic progression with the geometric one and gives general rule for the summation of any infinitesimal decreasing geometric progression. The formula for summing an infinitely decreasing progression was known to P. Fermat and other mathematicians of the 17th century.

Problems for arithmetic (and geometric) progressions are also found in the ancient Chinese tract "Mathematics in Nine Books", which, however, does not contain any instructions on the use of any summation formula.

The first progression problems that have come down to us are connected with the demands of economic life and social practice, such as the distribution of products, the division of inheritance, and so on.

From one cuneiform tablet, we can conclude that, observing the moon from new moon to full moon, the Babylonians came to the following conclusion: in the first five days after the new moon, the increase in illumination of the lunar disk occurs according to the law of geometric progression with a denominator of 2. In another later tablet, we are talking about summation geometric progression:

1+2+ +…+ . solution and answer S=512+(512-1), the data in the plate suggest that the author used the formula.

Sn= +( -1), but no one knows how he reached it.

The summation of geometric progressions and the compilation of corresponding problems that do not always meet practical needs were practiced by many lovers of mathematics throughout the ancient and middle ages.

Number Sequence Properties

Numeric sequence - a special case numeric function, and therefore some properties of functions (boundedness, monotonicity) are also considered for sequences.

Limited sequences

Sequence () is called bounded from above, that for any number n, M.

Sequence () is called bounded from below, if there is such a number m, that for any number n, m.

Sequence () is called bounded , if it is bounded from above and bounded from below, that is, there exists such a number M0 , which for any number n , M.

Sequence () is called unbounded , if there exists such a number M0 that there exists a number n such that, M.

Task: explore the sequence = to limitation.

Solution. The given sequence is bounded, since for any natural number n the following inequalities hold:

0 1,

That is, the sequence is bounded from below by zero, and at the same time is bounded from above by unity, and therefore is also bounded.

Answer: the sequence is limited - from below by zero, and from above by one.

Increasing and descending sequences

Sequence () is called increasing , if each term is greater than the previous one:

For example, 1, 3, 5, 7.....2n -1,... is an increasing sequence.

Sequence () is called decreasing , if each term is less than the previous one:

For example, 1; is a descending sequence.

Increasing and decreasing sequences are combined by a common term -monotonic sequences. Let's take a few more examples.

1; - this sequence is neither increasing nor decreasing (nonmonotonic sequence).

2n. It's about about the sequence 2, 4, 8, 16, 32, ... - increasing sequence.

In general, if a > 1, then the sequence= increases;

if 0 = decreasing.

Arithmetic progression

A numerical sequence, each member of which, starting from the second, is equal to the sum of the previous member and the same number d, is calledarithmetic progression, and the number d is the difference of an arithmetic progression.

So an arithmetic progression is numerical sequence

X, == + d, (n = 2, 3, 4, …; a and d are given numbers).

Example 1. 1, 3, 5, 7, 9, 11, ... is an increasing arithmetic progression, in which= 1, d = 2.

Example 2. 20, 17, 14, 11, 8, 5, 2, -1, -4, ... - a decreasing arithmetic progression, in which= 20, d = –3.

Example 3. Consider a sequence of natural numbers that, when divided by four, have a remainder of 1: 1; 5; 9; thirteen; 17; 21…

Each term, starting from the second, is obtained by adding the number 4 to the previous term. This sequence is an example of an arithmetic progression.

It is easy to find an explicit (formula) expressionthrough n. The value of the next element increases by d compared to the previous one, thus, the value of the n element will increase by (n - 1)d compared to the first member of the arithmetic progression, i.e.

= + d (n – 1). This is the formula for the nth term of an arithmetic progression.

This is the sum formula n members of an arithmetic progression.

The arithmetic progression is named because in it each term, except for the first, is equal to the arithmetic mean of the two adjacent to it - the previous and the next, indeed,

Geometric progression

A numerical sequence, all members of which 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, is calledgeometric progression, and the number q is the denominator of a geometric progression. Thus, a geometric progression is a numerical sequence (given recursively by the relations

B, = q (n = 2, 3, 4…; b and q are given numbers).

Example 1. 2, 6, 18, 54, ... - increasing geometric progression

2, q = 3.

Example 2. 2, -2, 2, -2, ... is a geometric progression= 2, q = –1.

One of the obvious properties of a geometric progression is that if a sequence is a geometric progression, then the sequence of squares, i.e.; ;…-

is a geometric progression whose first term is equal to, and the denominator is.

The formula for the nth member of a geometric progression is:

The formula for the sum of n members of a geometric progression:

characteristic propertygeometric 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 previous and subsequent members,

Conclusion

Numerical sequences have been studied by many scientists for many centuries.The first progression problems that have come down to us are connected with the demands of economic life and social practice, such as the distribution of products, the division of inheritance, and so on. They are one of the key concepts of mathematics. In my work, I tried to reflect the basic concepts associated with numerical sequences, how to set them, properties, and considered some of them. Separately, progressions (arithmetic and geometric) were considered, and the basic concepts associated with them were described.

Bibliography

1. A.G. Mordkovich, Algebra, Grade 10, textbook, 2012
2. A.G. Mordkovich, Algebra, grade 9, textbook, 2012
3. Great student guide. Moscow, "Drofa", 2001
4. G.I. Glaser, History of Mathematics in the School,

M.: Enlightenment, 1964.

1. "Mathematics at school", magazine, 2002.
2. Educational online services Webmath.ru
3. Universal popular science online encyclopedia "Krugosvet"

Before we start to decide arithmetic progression problems, consider what a number sequence is, since an arithmetic progression is a special case of a number sequence.

A numerical sequence is a numerical set, each element of which has its own serial number. The elements of this set are called members of the sequence. The ordinal number of a sequence element is indicated by an index:

The first element of the sequence;

The fifth element of the sequence;

- "nth" element of the sequence, i.e. the element "standing in the queue" at number n.

There is a dependency between the value of a sequence element and its ordinal number. Therefore, we can consider a sequence as a function whose argument is the ordinal number of an element of the sequence. In other words, one can say that the sequence is a function of the natural argument:

The sequence can be specified in three ways:

1 . The sequence can be specified using a table. In this case, we simply set the value of each member of the sequence.

For example, Someone decided to do personal time management, and to begin with, to calculate how much time he spends on VKontakte during the week. By writing the time in a table, he will get a sequence consisting of seven elements:

The first line of the table contains the number of the day of the week, the second - the time in minutes. We see that, that is, on Monday Someone spent 125 minutes on VKontakte, that is, on Thursday - 248 minutes, and, that is, on Friday, only 15.

2 . The sequence can be specified using the nth member formula.

In this case, the dependence of the value of a sequence element on its number is expressed directly as a formula.

For example, if , then

To find the value of a sequence element with a given number, we substitute the element number into the formula for the nth member.

We do the same if we need to find the value of a function if the value of the argument is known. We substitute the value of the argument instead in the equation of the function:

If, for example, , then

Once again, I note that in a sequence, in contrast to an arbitrary numeric function, the argument can only be natural number.

3 . The sequence can be specified using a formula that expresses the dependence of the value of the member of the sequence with number n on the value of the previous members. In this case, it is not enough for us to know only the number of a sequence member in order to find its value. We need to specify the first member or first few members of the sequence.

For example, consider the sequence ,

We can find the values ​​of the members of a sequence in sequence, starting from the third:

That is, each time to find the value of the nth member of the sequence, we return to the previous two. This way of sequencing is called recurrent, from the Latin word recurro- come back.

Now we can define an arithmetic progression. An arithmetic progression is a simple special case of a numerical sequence.

Arithmetic progression is called a numerical sequence, each member of which, starting from the second, is equal to the previous one, added with the same number.

The number is called the difference of an arithmetic progression. The difference of an arithmetic progression can be positive, negative, or zero.

If title="(!LANG:d>0">, то каждый член арифметической прогрессии больше предыдущего, и прогрессия является !} increasing.

For example, 2; 5; eight; eleven;...

If , then each term of the arithmetic progression is less than the previous one, and the progression is waning.

For example, 2; -one; -4; -7;...

If , then all members of the progression are equal to the same number, and the progression is stationary.

For example, 2;2;2;2;...

The main property of an arithmetic progression:

Let's look at the picture.

We see that

, and at the same time

Adding these two equalities, we get:

.

Divide both sides of the equation by 2:

So, each member of the arithmetic progression, starting from the second, is equal to the arithmetic mean of two neighboring ones:

Moreover, since

, and at the same time

, then

, and hence

Each member of the arithmetic progression starting with title="(!LANG:k>l">, равен среднему арифметическому двух равноотстоящих. !}

th member formula.

We see that for the members of the arithmetic progression, the following relations hold:

and finally

We got formula of the nth term.

IMPORTANT! Any member of an arithmetic progression can be expressed in terms of and . Knowing the first term and the difference of an arithmetic progression, you can find any of its members.

The sum of n members of an arithmetic progression.

In an arbitrary arithmetic progression, the sums of terms equally spaced from the extreme ones are equal to each other:

Consider an arithmetic progression with n members. Let the sum of n members of this progression be equal to .

Arrange the terms of the progression first in ascending order of numbers, and then in descending order:

Let's pair it up:

The sum in each parenthesis is , the number of pairs is n.

We get:

So, the sum of n members of an arithmetic progression can be found using the formulas:

Consider solving arithmetic progression problems.

1 . The sequence is given by the formula of the nth member: . Prove that this sequence is an arithmetic progression.

Let us prove that the difference between two adjacent members of the sequence is equal to the same number.

We have obtained that the difference of two adjacent members of the sequence does not depend on their number and is a constant. Therefore, by definition, this sequence is an arithmetic progression.

2 . Given an arithmetic progression -31; -27;...

a) Find the 31 terms of the progression.

b) Determine if the number 41 is included in this progression.

a) We see that ;

Let's write down the formula for the nth term for our progression.

In general

In our case , That's why

We get:

b) Suppose the number 41 is a member of the sequence. Let's find his number. To do this, we solve the equation:

We got a natural value of n, therefore, yes, the number 41 is a member of the progression. If the found value of n were not a natural number, then we would answer that the number 41 is NOT a member of the progression.

3 . a) Between the numbers 2 and 8, insert 4 numbers so that they, together with the given numbers, form an arithmetic progression.

b) Find the sum of the terms of the resulting progression.

a) Let's insert four numbers between the numbers 2 and 8:

We got an arithmetic progression, in which there are 6 terms.

Let's find the difference of this progression. To do this, we use the formula for the nth term:

Now it's easy to find the values ​​of the numbers:

3,2; 4,4; 5,6; 6,8

b)

Answer: a) yes; b) 30

4. The truck transports a batch of crushed stone weighing 240 tons, daily increasing the transportation rate by the same number of tons. It is known that 2 tons of rubble were transported on the first day. Determine how many tons of crushed stone were transported on the twelfth day if all the work was completed in 15 days.

According to the condition of the problem, the amount of crushed stone that the truck transports increases every day by the same number. Therefore, we are dealing with an arithmetic progression.

We formulate this problem in terms of an arithmetic progression.

During the first day, 2 tons of crushed stone were transported: a_1=2.

All work was completed in 15 days: .

The truck transports a batch of crushed stone weighing 240 tons:

We need to find .

First, let's find the progression difference. Let's use the formula for the sum of n members of the progression.

In our case:

The numerical 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 numerical sequence, in accordance with one of the most common definitions, is a mathematical function, the basis of which is the set of natural numbers arranged according to one pattern or another.

There are several options for creating number sequences.

First, this function can be specified in the so-called "explicit" way, when there is a certain formula by which each of its members can be determined by simply substituting the ordinal number into the given sequence.

The second method is called "recursive". Its essence lies in the fact that the first few members of the numerical sequence are given, as well as a special recursive formula, with the help of which, knowing the previous member, you can find the next one.

Finally, the most general way of specifying sequences is the so-called when, without much difficulty, one can not only identify one or another term under a certain serial number, but also, knowing several consecutive terms, come to the general formula of this function.

The numerical sequence can be decreasing or increasing. In the first case, each subsequent term is less than the previous one, and in the second, on the contrary, it is greater.

Considering this topic, it is impossible not to touch on the issue of the limits of sequences. The limit of a sequence is such a number when for any quantity, including for an infinitesimal quantity, there is an ordinal number after which the deviation of successive members of the sequence from 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 rather interesting properties.

Firstly, any numerical sequence is an example of a mathematical function, therefore, those properties that are characteristic of functions can be safely applied to sequences. The most striking example of such properties is the provision on increasing and decreasing arithmetic series, which are combined by one general concept are monotonic sequences.

Secondly, there is a fairly large group of sequences that cannot be classified as either increasing or decreasing - these are periodic sequences. In mathematics, they are considered to be those functions in which there is a so-called period length, that is, from a certain moment (n), the following equality begins to operate y n \u003d y n + T, where T will be the very length of the period.

Numeric sequence.
How ?

In this lesson, we will learn a lot of interesting things from the life of members of a large community called Vkontakte number sequences. The topic under consideration refers not only to the course of mathematical analysis, but also touches on the basics discrete mathematics. In addition, the material will be required for the development of other sections of the tower, in particular, during the study number series and functional rows. You can tritely say that this is important, you can say reassuringly that it’s simple, you can say a lot more on-duty phrases, but today is the first, unusually lazy school week, so it’s terribly breaking for me to write the first paragraph =) I already saved the file in my heart and got ready to sleep, suddenly… the idea of ​​a frank confession lit up the head, which incredibly relieved the soul and pushed for further tapping of the fingers on the keyboard.

Let's digress from summer memories and look into this fascinating and positive world of a new social network:

The concept of a numerical sequence

First, let's think about the word itself: what is a sequence? Consistency is when something is located behind something. For example, the sequence of actions, the sequence of the seasons. Or when someone is located behind someone. For example, a sequence of people in a queue, a sequence of elephants on a path to a watering hole.

Let's clarify immediately characteristics sequences. Firstly, sequence members are located strictly in a certain order. So, if two people in the queue are swapped, then this will already be another sequence. Secondly, to each sequence member you can assign a serial number:

It's the same with numbers. Let to each natural value according to some rule mapped to a real number. Then we say that a numerical sequence is given.

Yes, in mathematical problems unlike life situations, the sequence almost always contains infinitely many numbers.

Wherein:
called first member sequences;
– second member sequences;
– third member sequences;
…
– nth or common member sequences;
…

In practice, the sequence is usually given common term formula, For example:
is a sequence of positive even numbers:

Thus, the record uniquely determines all members of the sequence - this is the rule (formula) according to which the natural values numbers are matched. Therefore, the sequence is often briefly denoted by a common member, and other Latin letters can be used instead of "x", for example:

Sequence of positive odd numbers:

Another common sequence:

As, probably, many have noticed, the variable "en" plays the role of a kind of counter.

In fact, we dealt with numerical sequences back in middle school. Let's remember arithmetic progression. I will not rewrite the definition, let's touch on the essence of specific example. Let be the first term and step arithmetic progression. Then:
is the second term of this progression;
is the third member of this progression;
- fourth;
- fifth;
…
And, obviously, the nth member is asked recurrent formula

Note : in a recursive formula, each next term is expressed in terms of the previous term or even in terms of a whole set of previous terms.

The resulting formula is of little use in practice - to get, say, to , you need to go through all the previous terms. And in mathematics, a more convenient expression for the nth term of an arithmetic progression is derived: . In our case:

Substitute natural numbers in the formula and check the correctness of the numerical sequence constructed above.

Similar calculations can be made for geometric progression, the nth term of which is given by the formula , where is the first term , and is denominator progressions. In matan assignments, the first term is often equal to one.

progression sets the sequence ;
progression sets the sequence ;
progression sets the sequence ;
progression sets the sequence .

I hope everyone knows that -1 to an odd power is -1, and to an even power is one.

The progression is called infinitely decreasing, if (last two cases).

Let's add two new friends to our list, one of which has just knocked on the monitor matrix:

The sequence in mathematical jargon is called a "flasher":

In this way, sequence members can be repeated. So, in the considered example, the sequence consists of two infinitely alternating numbers.

Does it happen that the sequence consists of the same numbers? Certainly. For example, it sets an infinite number of "triples". For aesthetes, there is a case when “en” still formally appears in the formula:

Let's invite a simple girlfriend to dance:

What happens when "en" increases to infinity? Obviously, the terms of the sequence will infinitely close approach zero. This is the limit of this sequence, which is written as follows:

If the limit of the sequence zero, then it is called infinitesimal.

In the theory of mathematical analysis, it is given strict definition of the sequence limit through the so-called epsilon neighborhood. The next article will be devoted to this definition, but for now let's analyze its meaning:

Let us depict the terms of the sequence and the neighborhood symmetric with respect to zero (limit) on the real line:


Now hold the blue neighborhood with the edges of your palms and start to reduce it, pulling it to the limit (red dot). A number is the limit of a sequence if FOR ANY pre-selected -neighborhood (arbitrarily small) inside it will be infinitely many members of the sequence, and OUTSIDE of it - only final number of members (or none at all). That is, the epsilon neighborhood can be microscopic, and even less, but the “infinite tail” of the sequence must sooner or later fully enter this area.

The sequence is also infinitely small: with the difference that its members do not jump back and forth, but approach the limit exclusively from the right.

Naturally, the limit can be equal to any other finite number, an elementary example:

Here the fraction tends to zero, and accordingly, the limit is equal to "two".

If the sequence there is a finite limit, then it is called converging(in particular, infinitesimal at ). Otherwise - divergent, while two options are possible: either the limit does not exist at all, or it is infinite. In the latter case, the sequence is called infinitely large. Let's gallop through the examples of the first paragraph:

Sequences are infinitely large, as their members move steadily towards "plus infinity":

An arithmetic progression with the first term and a step is also infinitely large:

By the way, any arithmetic progression also diverges, except for the case with a zero step - when infinitely added to a specific number. The limit of such a sequence exists and coincides with the first term.

Sequences have a similar fate:

Any infinitely decreasing geometric progression, as the name implies, infinitely small:

If the denominator is a geometric progression, then the sequence is infinitely largeA:

If, for example, , then there is no limit at all, since the members tirelessly jump either to “plus infinity”, then to “minus infinity”. And common sense and matan's theorems suggest that if something strives somewhere, then this cherished place is unique.

After a little revelation it becomes clear that the flasher is to blame for the unrestrained throwing, which, by the way, diverges on its own.
Indeed, for a sequence it is easy to choose a -neighbourhood, which, say, clamps only the number -1. As a result, an infinite number of sequence members (“plus ones”) will remain outside the given neighborhood. But by definition, the "infinite tail" of the sequence from a certain moment (natural number) must fully enter ANY neighborhood of its limit. Conclusion: there is no limit.

Factorial is infinitely large sequence:

Moreover, it grows by leaps and bounds, so it is a number that has more than 100 digits (digits)! Why exactly 70? It asks for mercy my engineering calculator.

With a control shot, everything is a little more complicated, and we have just come to the practical part of the lecture, in which we will analyze combat examples:

But now it is necessary to be able to solve the limits of functions, at least at the level of two basic lessons: Limits. Solution examples and Remarkable Limits. Because many solution methods will be similar. But, first of all, let's analyze the fundamental differences between the limit of a sequence and the limit of a function:

In the limit of the sequence, the "dynamic" variable "en" can tend to only to "plus infinity"– in the direction of increasing natural numbers .
In the limit of the function, "x" can be directed anywhere - to "plus / minus infinity" or to an arbitrary real number.

Sequence discrete(discontinuous), that is, it consists of separate isolated members. One, two, three, four, five, the bunny went out for a walk. The argument of the function is characterized by continuity, that is, “x” smoothly, without incident, tends to one or another value. And, accordingly, the values ​​of the function will also continuously approach their limit.

Because of discreteness within the sequences there are their own branded things, such as factorials, flashers, progressions, etc. And now I will try to analyze the limits that are characteristic of sequences.

Let's start with progressions:

Example 1

Find the limit of a sequence

Solution: something similar to an infinitely decreasing geometric progression, but is it really? For clarity, we write out the first few terms:

Since , we are talking about sum members of an infinitely decreasing geometric progression, which is calculated by the formula .

Making a decision:

We use the formula for the sum of an infinitely decreasing geometric progression: . In this case: - the first term, - the denominator of the progression.

Example 2

Write the first four terms of the sequence and find its limit

This is an example for independent decision. To eliminate the uncertainty in the numerator, you will need to apply the formula for the sum of the first terms of an arithmetic progression:
, where is the first and is the nth term of the progression.

Since 'en' always tends to 'plus infinity' within sequences, it's not surprising that indeterminacy is one of the most popular.
And many examples are solved in exactly the same way as the limits of functions!

Or maybe something more complicated like ? Check out Example #3 of the article Limit Solving Methods.

From a formal point of view, the difference will be only in one letter - there is “x”, and here “en”.
The reception is the same - the numerator and denominator must be divided by "en" in the highest degree.

Also, within sequences, uncertainty is quite common. How to solve limits like can be found in Examples No. 11-13 of the same article.

To deal with the limit, refer to Example #7 of the lesson Remarkable Limits(second wonderful limit also valid for the discrete case). The solution will again be like a carbon copy with a difference in a single letter.

The following four examples (Nos. 3-6) are also “two-faced”, but in practice, for some reason, they are more typical for the limits of sequences than for the limits of functions:

Example 3

Find the limit of a sequence

Solution: first complete solution, then step by step comments:

(1) In the numerator we use the formula twice.

(2) We give like terms in the numerator.

(3) To eliminate uncertainty, we divide the numerator and denominator by ("en" in the highest degree).

As you can see, nothing complicated.

Example 4

Find the limit of a sequence

This is an example for a do-it-yourself solution, abbreviated multiplication formulas to help.

Within s demonstrative sequences applied similar method dividing the numerator and denominator:

Example 5

Find the limit of a sequence

Solution let's do it the same way:

A similar theorem is also true, by the way, for functions: the product of a bounded function by an infinitesimal function is an infinitesimal function.

Example 9

Find the limit of a sequence

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. didactic materials, in the problems of mathematical Olympiads, entrance exams to the Higher Schools and on the exam. I am interested to know the connection of mathematical sequences with other fields of knowledge.

Target research work: Expand knowledge of 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 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 \u003d ( x n ), if for an arbitrary predetermined arbitrarily small positive numberε 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 \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.

Thus, an arithmetic progression is a 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 instance:

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.