Mathematics is the science that builds the world. Both the scientist and the common man - no one can do without it. First, young children are taught to count, then add, subtract, multiply and divide; by middle school, letter symbols come into play, and in high school they can no longer be avoided.

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. In simpler words, this 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 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, using this method it is difficult to reach large values, for example up to a 125. . Especially for such cases, a formula convenient for practice was derived: a n =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 types of 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.

Determining the Sequence Limit

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:

1. All limits are abbreviated as lim.
2. The notation of a limit consists of the abbreviation lim, any variable tending to a certain number, zero or infinity, as well as the function itself.

It is easy to understand that the definition of the limit of a sequence can be formulated as follows: this 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.

General designation for the limit of sequences

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.

Uncertainty and certainty of the limit

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.

What is a neighborhood?

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 included 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

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 solution or proof much easier:

1. Uniqueness of the limit of a sequence. Any sequence can have only one limit or none at all. The same example with a queue that can only have one end.
2. If a series of numbers has a limit, then the sequence of these numbers is limited.
3. The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
4. The limit of the quotient of dividing two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.

Proof of sequences

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.

Or maybe he's not there?

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.

Monotonic sequence

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.

Limit of a convergent and bounded 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.

Limit of a monotonic 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!

Various actions with limits

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 is equal to the product of their limits. The same is true for division: the limit of the quotient of two sequences is equal to the quotient of their limits, provided that the limit is not zero. After all, if the limit of sequences is equal to zero, then division by zero will result, which is impossible.

Properties of sequence quantities

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:

1. The sum of any number of any number of small quantities will also be a small quantity.
2. The sum of any number of large quantities will be an infinitely large quantity.
3. The product of arbitrarily small quantities is infinitesimal.
4. The product of any number of large numbers is infinitely large.
5. If the original sequence tends to an infinitely large number, then its inverse will be infinitesimal and tend to zero.

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.

Number 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 relates not only to the course of mathematical analysis, but also touches on the basics discrete mathematics. In addition, the material will be required for mastering other sections of the tower, in particular, during the study number series And functional series. You can say tritely that this is important, you can say encouragingly that it’s simple, you can say many more routine phrases, but today is the first, unusually lazy week of school, so it’s terribly breaking me to write the first paragraph =) I’ve already saved the file in my hearts and got ready to sleep, when suddenly... my head was illuminated by the idea of ​​a sincere confession, which incredibly lightened my soul and pushed me to continue tapping my fingers on the keyboard.

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

Concept of number sequence

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

Let us immediately clarify the characteristic features of the sequence. Firstly, sequence members are located strictly in a certain order. So, if two people in the queue are swapped, then this will already be other subsequence. Secondly, everyone sequence member You can assign a serial number:

It's the same with numbers. Let to each natural value according to some rule matched to a real number. Then they 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:
– sequence of positive even numbers:

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

Sequence of positive odd numbers:

Another common sequence:

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

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

Note : in a recurrent formula, each subsequent 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 has been derived: . In our case:

Substitute natural numbers into 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 – denominator progression. In math tasks, 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 equal to –1, and to an even power – one.

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

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

The sequence in mathematical jargon is called a “blinker”:

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

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

Let's invite a simple friend to dance:

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

If the limit of a sequence is 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 look at its meaning:

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


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

The sequence is also infinitesimal: 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 convergent(in particular, infinitesimal at ). Otherwise - divergent, in this case, 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 confidently move towards “plus infinity”:

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

By the way, any arithmetic progression also diverges, with the exception of the case with a zero step - when . The limit of such a sequence exists and coincides with the first term.

The sequences have a similar fate:

Any infinitely decreasing geometric progression, as is clear from the name, infinitely small:

If the denominator of the geometric progression is , then the sequence is infinitely large:

If, for example, then the limit does not exist at all, since the members tirelessly jump either to “plus infinity” or to “minus infinity”. And common sense and Matan’s theorems suggest that if something is striving somewhere, then this is the only cherished place.

After a little revelation it becomes clear that the “flashing light” is to blame for the uncontrollable throwing, which, by the way, diverges on its own.
Indeed, for a sequence it is easy to choose a -neighborhood that, say, only clamps the number –1. As a result, an infinite number of sequence members (“plus ones”) will remain outside this neighborhood. But by definition, the “infinite tail” of the sequence from a certain moment (natural number) must fully go into ANY vicinity of your limit. Conclusion: the sky is the limit.

Factorial is infinitely large sequence:

Moreover, it is growing by leaps and bounds, so it is a number that has more than 100 digits (digits)! Why exactly 70? On it my engineering microcalculator begs for mercy.

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 you need to be able to solve the limits of functions, at least at the level of two basic lessons: Limits. Examples of solutions And Wonderful 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”– towards increasing natural numbers .
In the limit of the function, “x” can be directed anywhere – to “plus/minus infinity” or to an arbitrary real number.

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

Because of discreteness within the sequences there are their own signature things, such as factorials, “flashing lights”, progressions, etc. And now I will try to analyze the limits that are specific to sequences.

Let's start with progressions:

Example 1

Find the limit of the sequence

Solution: something similar to an infinitely decreasing geometric progression, but is it really that? For clarity, let’s write down the first few terms:

Since, then we are talking about amount terms of an infinitely decreasing geometric progression, which is calculated by the formula.

Let's make 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 you to solve on your own. 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 a is the nth term of the progression.

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

Or maybe something more complicated like ? Check out Example No. 3 of the article Methods for solving limits.

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

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

To understand the limit, refer to Example No. 7 of the lesson Wonderful Limits(the second remarkable limit is also valid for the discrete case). The solution will again be like a carbon copy with a single letter difference.

The next four examples (Nos. 3-6) are also “two-faced”, but in practice for some reason they are more characteristic of sequence limits than of function limits:

Example 3

Find the limit of the sequence

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

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

(2) We present similar terms in the numerator.

(3) To eliminate uncertainty, divide the numerator and denominator by (“en” to the highest degree).

As you can see, nothing complicated.

Example 4

Find the limit of the sequence

This is an example for you to solve on your own, abbreviated multiplication formulas to help.

Within s indicative Sequences use a similar method of dividing the numerator and denominator:

Example 5

Find the limit of the sequence

Solution Let's arrange it according to the same scheme:

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

Example 9

Find the limit of the sequence

The function a n =f (n) of the natural argument n (n=1; 2; 3; 4;...) is called a number sequence.

Numbers a 1; a 2 ; a 3 ; a 4 ;…, forming a sequence, are called members of a numerical sequence. So a 1 =f (1); a 2 =f (2); a 3 =f (3); a 4 =f (4);…

So, the members of the sequence are designated by letters indicating the indices - the serial numbers of their members: a 1 ; a 2 ; a 3 ; a 4 ;…, therefore, a 1 is the first member of the sequence;

a 2 is the second term of the sequence;

a 3 is the third member of the sequence;

a 4 is the fourth term of the sequence, etc.

Briefly the numerical sequence is written as follows: a n =f (n) or (a n).

There are the following ways to specify a number sequence:

1) Verbal method. Represents a pattern or rule for the arrangement of members of a sequence, described in words.

Example 1. Write a sequence of all non-negative numbers that are multiples of 5.

Solution. Since all numbers ending in 0 or 5 are divisible by 5, the sequence will be written like this:

0; 5; 10; 15; 20; 25; ...

Example 2. Given the sequence: 1; 4; 9; 16; 25; 36; ... . Ask it verbally.

Solution. We notice that 1=1 2 ; 4=2 2 ; 9=3 2 ; 16=4 2 ; 25=5 2 ; 36=6 2 ; ... We conclude: given a sequence consisting of squares of natural numbers.

2) Analytical method. The sequence is given by the formula of the nth term: a n =f (n). Using this formula, you can find any member of the sequence.

Example 3. The expression for the kth term of a number sequence is known: a k = 3+2·(k+1). Compute the first four terms of this sequence.

a 1 =3+2∙(1+1)=3+4=7;

a 2 =3+2∙(2+1)=3+6=9;

a 3 =3+2∙(3+1)=3+8=11;

a 4 =3+2∙(4+1)=3+10=13.

Example 4. Determine the rule for composing a numerical sequence using its first few members and express the general term of the sequence using a simpler formula: 1; 3; 5; 7; 9; ... .

Solution. We notice that we are given a sequence of odd numbers. Any odd number can be written in the form: 2k-1, where k is a natural number, i.e. k=1; 2; 3; 4; ... . Answer: a k =2k-1.

3) Recurrent method. The sequence is also given by a formula, but not by a general term formula, which depends only on the number of the term. A formula is specified by which each next term is found through the previous terms. In the case of the recurrent method of specifying a function, one or several first members of the sequence are always additionally specified.

Example 5. Write out the first four terms of the sequence (a n ),

if a 1 =7; a n+1 = 5+a n .

a 2 =5+a 1 =5+7=12;

a 3 =5+a 2 =5+12=17;

a 4 =5+a 3 =5+17=22. Answer: 7; 12; 17; 22; ... .

Example 6. Write out the first five terms of the sequence (b n),

if b 1 = -2, b 2 = 3; b n+2 = 2b n +b n+1 .

b 3 = 2∙b 1 + b 2 = 2∙(-2) + 3 = -4+3=-1;

b 4 = 2∙b 2 + b 3 = 2∙3 +(-1) = 6 -1 = 5;

b 5 = 2∙b 3 + b 4 = 2∙(-1) + 5 = -2 +5 = 3. Answer: -2; 3; -1; 5; 3; ... .

4) Graphic method. The numerical sequence is given by a graph, which represents isolated points. The abscissas of these points are natural numbers: n=1; 2; 3; 4; ... . Ordinates are the values ​​of the sequence members: a 1 ; a 2 ; a 3 ; a 4 ;… .

Example 7. Write down all five terms of the numerical sequence given graphically.

Each point in this coordinate plane has coordinates (n; a n). Let's write down the coordinates of the marked points in ascending order of the abscissa n.

We get: (1 ; -3), (2 ; 1), (3 ; 4), (4 ; 6), (5 ; 7).

Therefore, a 1 = -3; a 2 =1; a 3 =4; a 4 =6; a 5 =7.

Answer: -3; 1; 4; 6; 7.

The considered numerical sequence as a function (in example 7) is given on the set of the first five natural numbers (n=1; 2; 3; 4; 5), therefore, is finite number sequence(consists of five members).

If a number sequence as a function is given on the entire set of natural numbers, then such a sequence will be an infinite number sequence.

The number sequence is called increasing, if its members are increasing (a n+1 >a n) and decreasing, if its members are decreasing(a n+1

An increasing or decreasing number sequence is called monotonous.

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.

Properties of number sequences.

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.

Geometric progression.

A numerical sequence, all of whose terms are non-zero and each of whose terms, 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.

Consistency limit.

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 an 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 Chugainova

Oganesyan Eva

Number sequences. Abstract.

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Municipal budgetary educational institution
"Secondary school No. 31"
city ​​of Barnaul

Number sequences

Essay

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

Barnaul - 2014

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

Number sequences.……………………………………………………………...3

Methods for specifying number sequences………………………...4

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

Properties of number sequences…………………………………7

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

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

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

References………………………………………………………11

Introduction

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

1. Study the historical aspects of the development of the doctrine of progressions;
2. Consider methods of specifying and properties of number sequences;
3. Get acquainted with arithmetic and geometric progressions.

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

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

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

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

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

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

1, ½, 1 /3, ¼, 1 /5,… - a sequence of numbers that are reciprocal to the natural numbers.

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

Progressions - particular types of numerical sequences - are found in monuments of the 2nd millennium BC. e.

Number sequences

There are various definitions of a number sequence.

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

Number sequence – it is a numbered number set.

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

, , , …, To denote the sequence, 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, etc., so we get the sequence: 2; 4; 6; 8; 10 ….

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

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

; ; ; ; ; … .

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

The numbers that form the sequence are called first, second, third, fourth, etc., respectively. members of the sequence. Members of a sequence are usually designated by letters with indices indicating the serial number of the member. For example:, , etc. in general, the member of the sequence with number n, or, as they say, the nth member of the sequence, denotes. The sequence itself is denoted by (). A sequence can contain either an infinite number of terms or a finite number. In this case it is called final. For example: a sequence of two-digit numbers.10; eleven; 12; 13; ...; 98; 99

Methods for specifying number sequences

Sequences can be specified in several ways.

Usually it is more appropriate to set the sequencethe formula for its common nth term, which allows you to find any member of the sequence knowing its number. In this case we say that the sequence is given analytically. For example: sequence of positive even terms=2n.

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

6; 20; 56; 144; 352;…

Solution. Let's write each member 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 we can see, the terms of the sequence are the product of a power of two multiplied by successive odd numbers, with two raised to a power that is equal to the number of the element in question. Thus, we conclude that

Answer: general term formula:

Another way to specify a sequence is to specify the sequence usingrecurrence relation. A formula expressing any member of a sequence, starting from 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 recurrently 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 specified recurrently:

N N, = 1.

Task: subsequenceis given using 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 specifying sequences recurrently, calculations turn out to be 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 previous 499 members.

Descriptive methodassignment of a number sequence is that it explains 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.

The numerical sequence can also be specified simplylisting its members.

Development of the doctrine of progressions

The word progression is of Latin origin (progressio), literally means “movement forward” (like the word “progress”) and is found for the first time in the Roman author Boethius (V-VI century). Initially, progression was understood as any numerical sequence constructed according to a law that allows 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 in common use. In the 17th century, for example, J. Gregory uses the term “series” instead of progression, and another prominent English mathematician, J. Wallis, uses the term “infinite progressions” for infinite series.

Currently we consider progressions as special cases of number sequences.

Theoretical information related to progressions is first found in the documents of Ancient Greece that have reached us.

In the Psammit, Archimedes first compares the 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, arithmetic progression began to be denoted this way.

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

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

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

Some formulas related to progressions were known to Chinese and Indian scientists. Thus, Aryabhatta (5th century) knew formulas for the general term, the sum of an arithmetic progression, etc., Magavira (9th 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 Abacus (1202) by Leonardo of Pisa. In “The Science of Numbers” (1484), N. Schuke, like Archimedes, compares the arithmetic progression with the geometric one and gives a 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 on 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 related to the demands of economic life and social practice, such as the distribution of products, division of inheritance, etc.

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 the summation geometric progression:

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

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

The summation of geometric progressions and the compilation of corresponding problems, which did not always meet practical needs, were carried out by many amateurs of mathematics throughout the ancient and Middle Ages.

Properties of number sequences

A numerical sequence is a special case of a numerical function, and therefore some properties of functions (boundedness, monotonicity) are also considered for sequences.

Restricted Sequences

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

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

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

Subsequence () is called unlimited , if there is such a number M0 that there is a number n such that, M.

Task: explore sequence = to limitations.

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

0 1,

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

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

Ascending and descending sequences

Subsequence () is called increasing , if each member is greater than the previous one:

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

Subsequence () is called decreasing , if each of its members is less than the previous one:

For example, 1; - decreasing sequence.

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

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

2n. We are talking about the sequence 2, 4, 8, 16, 32, ... - an increasing sequence.

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

if 0 = decreases.

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 the arithmetic progression.

Thus, an arithmetic progression is a number 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, which= 1, d = 2.

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

Example 3. Consider a sequence of natural numbers that, when divided by four, give a remainder of 1: 1; 5; 9; 13; 17; 21…

Each of its terms, 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 not difficult to find an explicit (formular) expressionthrough n. The value of the next element increases by d compared to the previous one, thus, the value n of the element will increase by (n – 1)d compared to the first term 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 terms of an arithmetic progression.

The progression is called an arithmetic progression because each term in it, except the first, is equal to the arithmetic mean of the two adjacent to it - the previous and the subsequent, indeed,

Geometric progression

A numerical sequence, all of whose terms are different from zero and each of whose terms, starting from the second, is obtained from the previous term by multiplying by the same number q, is calledgeometric progression, and the number q is the denominator of the geometric progression. Thus, a geometric progression is a number 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, … – geometric progression= 2, q = –1.

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.; ;…-

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

The formula for the nth term of the geometric progression is:

Formula for the sum of n terms 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 of the previous and subsequent terms,

Conclusion

Many scientists have been studying number sequences for many centuries.The first progression problems that have come down to us are related to the demands of economic life and social practice, such as the distribution of products, division of inheritance, etc. They are one of the key concepts of mathematics. In my work, I tried to reflect the basic concepts associated with numerical sequences, methods of defining them, properties, and considered some of them. Separately, progressions (arithmetic and geometric) were considered and the basic concepts associated with them were discussed.

Bibliography

1. A.G. Mordkovich, Algebra, 10th grade, textbook, 2012.
2. A.G. Mordkovich, Algebra, 9th grade, textbook, 2012.
3. Great reference book for schoolchildren. Moscow, Bustard, 2001.
4. G.I. Glaser, “The History of Mathematics in School,”

M.: Education, 1964.

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