Numeric sequences classification of sequences limit of a numerical sequence. How to calculate the limits of sequences? Determining the Sequence Limit

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 indices - serial numbers 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, 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 - 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.

Every 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.

Reviewed number 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.

This material is presented in the section Limit of a sequence - basic theorems and properties. Here we will look at some examples of sequences.
.
Sequence Examples Examples of infinitely increasing sequences Consider the sequence.

The common member of this sequence is . Let's write down the first few terms: It can be seen that as the number n increases, the elements increase indefinitely towards
.
positive values . We can say that this sequence tends to: for . Now consider the sequence with

common member

.
.
Here are its first few members: = 0 As the number n increases, the elements of this sequence increase indefinitely in = 0 absolute value > 0 , but do not have a constant sign. That is, this sequence tends to: at .

Examples of sequences converging to a finite number
.
In this sequence, terms with even numbers are equal to zero. Terms with odd n are equal. = 0 Therefore, as n increases, their values ​​approach the limiting value a
.
. > 0 This also follows from the fact that = 0 Just like in the previous example, we can specify an arbitrarily small error ε = 0 , for which it is possible to find a number N such that elements with numbers greater than N will deviate from the limit value a

by an amount not exceeding the specified error. Therefore this sequence converges to the value a

: at .

Examples of divergent sequences


.
Consider a sequence with the following common term:
,
Here are its first members: 1 = 0 It can be seen that terms with even numbers:
,
Here are its first members: 2 = 2 converge to the value a

.

Odd-numbered members: . The sequence itself, as n grows, does not converge to any value.
.
Sequence with terms distributed in the interval (0;1)
.
Now let's look at a more interesting sequence. On numeric


.
let's take the straight line

line segment . Let's divide it in half. We get two segments. Let (0; 1) Let's divide each of the segments in half again. We get four segments. Let Let's divide each segment in half again. Let's take

And so on. As a result, we obtain a sequence whose elements are distributed in an open interval. Whatever point we take from the closed interval

, we can always find members of the sequence that will be arbitrarily close to this point or coincide with it. = 0 Then from the original sequence one can select a subsequence that will converge to
.
= 0 .

arbitrary point = 1 from the interval
.
. = 1 .

That is, as the number n increases, the members of the subsequence will come closer and closer to the pre-selected point. For example, for point a you can choose the following subsequence:

For point a

Let's choose the following subsequence: The terms of this subsequence converge to the value a Since there are subsequences converging to

different meanings
,
, then the original sequence itself does not converge to any number.
Sequence containing all rational numbers

To do this, draw the p and q axes on the plane. We draw grid lines through the integer values ​​of p and q. Then each node of this grid with will correspond (0; 0) rational number < 1 . The entire set of rational numbers will be represented by a set of nodes. We need to find a way to number all the nodes so that we don't miss any nodes. This is easy to do if you number the nodes by squares, the centers of which are located at the point

(see picture). In this case, the lower parts of the squares with q
.
we don't need it. Therefore they are not shown in the figure. So, for the top side of the first square we have: Next we number

.
top part

.
let's take the straight line

the following square:

We number the top part of the following square: In this way we obtain a sequence containing all rational numbers. You can notice that any rational number appears in this sequence an infinite number of times. Indeed, along with the node , this sequence will also include nodes , where is a natural number. But all these nodes correspond to the same rational number. Then from the sequence we have constructed, we can select a subsequence (having an infinite number of elements), all of whose elements are equal to a predetermined rational number. Since the sequence we constructed has subsequences converging to

different numbers

, then the sequence does not converge to any number. Conclusion Here we have given a precise definition of the number sequence. We also raised the issue of its convergence, based on intuitive ideas.

Precise definition convergence is discussed on the page Determining the Limit of a Sequence. Related properties and theorems are stated on the page For many people mathematical analysis is just a set of incomprehensible numbers, icons and definitions, far from real life. However, the world in which we exist is built on numerical patterns, the identification of which helps not only to cognize the world and solve it complex problems, but also to simplify household

practical problems

. What does a mathematician mean when he says that a number sequence converges? We should talk about this in more detail. small? such bright figures, the resulting row will turn out to be fantastically long. This is a convergent number sequence. And it tends to zero, since the size of each subsequent nesting doll, catastrophically decreasing, gradually turns into nothing. Thus, it is easy to explain what the infinitesimal is.

A similar example would be a road going into the distance. And the visual dimensions of the car driving away from the observer along it, gradually shrinking, turn into a shapeless speck resembling a point. Thus, the car, like some object, moving away in an unknown direction, becomes infinitely small. Options the specified body will never be zero in literally this word, but invariably strive for this value in the final limit. Therefore, this sequence converges again to zero.

Let's calculate everything drop by drop

Let's imagine now everyday situation. The doctor prescribed the patient to take the mixture, starting with ten drops per day and adding two drops every subsequent day. And so the doctor suggested continuing until the contents of the bottle of medicine, the volume of which is 190 drops, are gone. From the above it follows that the number of such, listed by day, will be the following number series: 10, 12, 14 and so on.

How to find out the time to complete the entire course and the number of members of the sequence? Here, of course, you can count the drops in a primitive way. But it is much easier, given the pattern, to use the formula with a step d = 2. And using this method, find out that the number of terms number series equals 10. Moreover, a 10 = 28. The member number indicates the number of days the medicine is taken, and 28 corresponds to the number of drops that the patient must take on the last day. Does this sequence converge? No, because, despite the fact that it is limited at the bottom by the number 10, and at the top - 28, such a number series has no limit, unlike the previous examples.

What is the difference?

Let us now try to clarify: when a number series turns out to be a convergent sequence. A definition of this kind, as can be concluded from the above, is directly related to the concept of a finite limit, the presence of which reveals the essence of the issue. So what? fundamental difference the previously given examples? And why in the last of them the number 28 cannot be considered the limit of the number series X n = 10 + 2(n-1)?

To clarify this question, consider another sequence given by the formula below, where n belongs to the set of natural numbers.

This community of members is a collection ordinary fractions, the numerator of which is 1, and the denominator is constantly increasing: 1, ½ ...

Moreover, each subsequent representative of this series is increasingly closer to 0 in location on the number line. This means that a neighborhood appears where the points cluster around zero, which is the limit. And the closer they are to it, the denser their concentration on the number line becomes. And the distance between them is catastrophically reduced, turning into infinitesimal. This is a sign that the sequence is convergent.

In the same way, the multi-colored rectangles depicted in the figure, when removed in space, are visually arranged more closely together, in the hypothetical limit turning into negligible ones.

Infinitely large sequences

Having examined the definition of a convergent sequence, we now move on to counter examples. Many of them have been known to man since ancient times. The simplest variants of divergent sequences are series of natural and even numbers. They are otherwise called infinitely large, since their members, constantly increasing, are increasingly approaching positive infinity.

Examples of these can also include any of the arithmetic and geometric progressions with step and denominator respectively Above zero. Divergent sequences are also considered to be numerical series that have no limit at all. For example, X n = (-2) n -1 .

Fibonacci sequence

The practical benefits of the previously mentioned number series for humanity are undeniable. But there are a huge number of others wonderful examples. One of them is the Fibonacci sequence. Each of its terms, which begin with one, is the sum of the previous ones. Its first two representatives are 1 and 1. The third is 1+1=2, the fourth is 1+2=3, the fifth is 2+3=5. Further, according to the same logic, follow the numbers 8, 13, 21 and so on.

This series of numbers increases indefinitely and has no finite limit. But he has one more remarkable property. The ratio of each previous number to the next one is increasingly approaching 0.618 in value. Here you can understand the difference between a convergent and divergent sequence, because if you compile a series of quotients obtained from divisions, the indicated numerical system will have final limit equal to 0.618.

Sequence of Fibonacci ratios

The above number series is widely used in practical purposes for technical analysis of markets. But this does not limit its capabilities, which the Egyptians and Greeks knew and were able to put into practice in ancient times. This is proven by the pyramids and the Parthenon they built. After all, the number 0.618 is constant coefficient the golden ratio, well known in ancient times. According to this rule, any arbitrary segment can be divided so that the relationship between its parts will coincide with the relationship between the largest of the segments and the total length.

Let's build a series of these relationships and try to analyze this sequence. The number series will be as follows: 1; 0.5; 0.67; 0.6; 0.625; 0.615; 0.619 and so on. Continuing in this way, we can verify that the limit of the convergent sequence will indeed be 0.618. However, it is necessary to note other properties of this pattern. Here the numbers seem to be out of order, and not at all in ascending or descending order. This means that this convergent sequence is not monotonic. Why this is so will be discussed further.

Monotony and limitation

Members of a number series with increasing numbers can clearly decrease (if x 1 >x 2 >x 3 >…>x n >…) or increase (if x 1

Having written down the numbers of this series, you can see that any of its members, indefinitely approaching 1, will never exceed this value. In this case, the convergent sequence is said to be bounded. This happens whenever there is something like this positive number M, which always turns out to be greater than any of the terms of the series in modulus. If a number series has signs of monotonicity and has a limit, and therefore converges, then it is necessarily endowed with this property. Moreover, the opposite does not have to be true. This is evidenced by the theorem on the boundedness of a convergent sequence.

The application of such observations in practice turns out to be very useful. Let's give a specific example, examining the properties of the sequence X n = n/n+1, and prove its convergence. It is easy to show that it is monotonic, since (x n +1 - x n) is a positive number for any value of n. The limit of the sequence is equal to the number 1, which means that all the conditions of the above theorem, also called Weierstrass’s theorem, are met. The boundedness theorem for a convergent sequence states that if it has a limit, then it is bounded in any case. However, let's give the following example. The number series X n = (-1) n is bounded below by the number -1 and above by 1. But this sequence is not monotonic, has no limit and therefore does not converge. That is, limitedness does not always imply the presence of a limit and convergence. For this to happen, the lower and upper limits must coincide, as in the case of the Fibonacci ratios.

Numbers and laws of the Universe

The simplest variants of a convergent and divergent sequence are, perhaps, the number series X n = n and X n = 1/n. The first of them is a natural series of numbers. It is, as already mentioned, infinitely large. The second convergent sequence is bounded, and its terms approach infinitesimal in magnitude. Each of these formulas personifies one of the sides of the multifaceted Universe, helping a person, in the language of numbers and signs, to imagine and calculate something unknowable, inaccessible to limited perception.

The laws of the universe, ranging from the insignificant to the incredibly large, are also expressed by the golden coefficient of 0.618. Scientists believe that it lies at the core of the essence of things and is used by nature to form its parts. The previously mentioned relationships between the subsequent and previous members of the Fibonacci series do not complete the demonstration of the amazing properties of this unique series. If we consider the quotient of dividing the previous term by the next one by one, we get the series 0.5; 0.33; 0.4; 0.375; 0.384; 0.380; 0.382 and so on. The interesting thing is that this limited sequence converges, it is not monotonic, but the ratio of adjacent numbers extreme from a certain term always turns out to be approximately equal to 0.382, which can also be used in architecture, technical analysis and other industries.

There are other interesting coefficients of the Fibonacci series, they all play a special role in nature, and are also used by humans for practical purposes. Mathematicians are confident that the Universe is developing along a kind of “golden spiral” formed from the indicated coefficients. With their help, it is possible to calculate many phenomena occurring on Earth and in space, from the growth of the number of certain bacteria to the movement of distant comets. As it turns out, the DNA code is subject to similar laws.

Decreasing geometric progression

There is a theorem stating the uniqueness of the limit of a convergent sequence. This means that it cannot have two or more limits, which is undoubtedly important for finding its mathematical characteristics.

Let's look at some cases. Any number series made up of members of an arithmetic progression is divergent, except for the case with zero step. The same applies to a geometric progression whose denominator is greater than 1. The limits of such number series are “plus” or “minus” of infinity. If the denominator is less than -1, then there is no limit at all. Other options are also possible.

Let's consider a number series given by the formula X n = (1/4) n -1. At first glance, it is easy to understand that this convergent sequence is bounded because it is strictly decreasing and in no way capable of taking negative values.

Let us write a certain number of its members in a series.

It turns out: 1; 0.25; 0.0625; 0.015625; 0.00390625 and so on. Quite simple calculations are enough to understand how quickly this geometric progression starts from denominators 0

Fundamental Sequences

Augustin Louis Cauchy, a French scientist, showed the world many works related to mathematical analysis. He gave definitions to such concepts as differential, integral, limit and continuity. He also investigated the basic properties of convergent sequences. In order to understand the essence of his ideas, it is necessary to summarize some important details.

At the very beginning of the article, it was shown that there are such sequences for which there is a neighborhood where the points representing the members of a certain series on the number line begin to crowd together, lining up more and more densely. At the same time, the distance between them decreases as the number of the next representative increases, turning into infinitesimal. Thus, it turns out that in a given neighborhood an infinite number of representatives of a given series are grouped, while outside it there is a finite number of them. Such sequences are called fundamental.

The famous Cauchy criterion, created by a French mathematician, clearly indicates that the presence of such a property is sufficient to prove that the sequence converges. The opposite is also true.

It should be noted that this conclusion of the French mathematician is for the most part of purely theoretical interest. Its application in practice is considered quite difficult, therefore, in order to determine the convergence of series, it is much more important to prove the existence of a finite limit for the sequence. Otherwise, it is considered divergent.

When solving problems, you should also take into account the basic properties of convergent sequences. They are presented below.

Infinite amounts

Such famous ancient scientists as Archimedes, Euclid, Eudoxus used sums of infinite number series to calculate the lengths of curves, volumes of bodies and areas of figures. In particular, this is how it was possible to find out the area of ​​a parabolic segment. For this purpose, the sum of the number series of a geometric progression with q = 1/4 was used. The volumes and areas of other arbitrary figures were found in a similar way. This option was called the “exhaustion” method. The idea was that the body being studied, complex in shape, was divided into parts, which represented figures with easily measurable parameters. For this reason, it was not difficult to calculate their areas and volumes, and then they were added up.

By the way, similar problems are very familiar to modern schoolchildren and are found in Unified State Examination tasks. A unique method, found by distant ancestors, is still the simplest solution today. Even if there are only two or three parts into which a numerical figure is divided, the addition of their areas still represents the sum of the number series.

Much later, the ancient Greek scientists Leibniz and Newton, based on the experience of their wise predecessors, learned the laws of integral calculation. Knowledge of the properties of sequences helped them solve differential and algebraic equations. Currently, the theory of series, created through the efforts of many generations of talented scientists, provides a chance to solve a huge number of mathematical and practical problems. And the study of numerical sequences is the main problem solved by mathematical analysis since its creation.

Consider a series of natural numbers: 1, 2, 3, , n – 1, n,  .

If we replace every natural number n in this series by a certain number a n, following some law, we get a new series of numbers:

a 1 , a 2 , a 3 , , a n –1 , a n , ,

briefly designated and called numerical sequence. Magnitude a n is called a common member of a number sequence. Usually the number sequence is given by some formula a n = f(n) allowing you to find any member of the sequence by its number n; this formula is called the general term formula. Note that it is not always possible to define a numerical sequence using a general term formula; sometimes a sequence is specified by describing its members.

By definition, a sequence always contains an infinite number of elements: any two different elements differ at least in their numbers, of which there are infinitely many.

A number sequence is a special case of a function. A sequence is a function defined on the set of natural numbers and taking values ​​in the set of real numbers, i.e. a function of the form f : N  R.

Subsequence
called increasing(decreasing), if for any n  N
Such sequences are called strictly monotonous.

Sometimes it is convenient to use not all natural numbers as numbers, but only some of them (for example, natural numbers starting from some natural number n 0). For numbering it is also possible to use not only natural numbers, but also other numbers, for example, n= 0, 1, 2,  (here zero is added as another number to the set of natural numbers). In such cases, when specifying the sequence, indicate what values ​​the numbers take n.

If in some sequence for any n  N
then the sequence is called non-decreasing(non-increasing). Such sequences are called monotonous.

Example 1 . The number sequence 1, 2, 3, 4, 5, ... is a series of natural numbers and has a common term a n = n.

Example 2 . The number sequence 2, 4, 6, 8, 10, ... is a series of even numbers and has a common term a n = 2n.

Example 3 . 1.4, 1.41, 1.414, 1.4142, … – a numerical sequence of approximate values ​​with increasing accuracy.

In the last example it is impossible to give a formula for the general term of the sequence.

Example 4 . Write the first 5 terms of a number sequence using its common term
. To calculate a 1 is needed in the formula for the general term a n instead of n substitute 1 to calculate a 2 − 2, etc. Then we have:

Test 6 . The common member of the sequence 1, 2, 6, 24, 120,  is:

1)

2)

3)

4)

Test 7 .
is:

1)

2)

3)

4)

Test 8 . Common member of the sequence
is:

1)

2)

3)

4)

Number sequence limit

Consider a number sequence whose common term approaches some number A when the serial number increases n. In this case, the number sequence is said to have a limit. This concept has a more strict definition.

Number A called the limit of a number sequence
:

(1)

if for any  > 0 there is such a number n 0 = n 0 (), depending on , which
at n > n 0 .

This definition means that A there is a limit to a number sequence if its common term approaches without limit A with increasing n. Geometrically, this means that for any  > 0 one can find such a number n 0 , which, starting from n > n 0 , all members of the sequence are located inside the interval ( A – , A+ ). A sequence having a limit is called convergent; otherwise - divergent.

A number sequence can have only one limit (finite or infinite) of a certain sign.

Example 5 . Harmonic sequence has the limit number 0. Indeed, for any interval (–; +) as a number N 0 can be any integer greater than . Then for everyone n > n 0 >we have

Example 6 . The sequence 2, 5, 2, 5,  is divergent. Indeed, no interval of length less than, for example, one, can contain all members of the sequence, starting from a certain number.

The sequence is called limited, if such a number exists M, What
for all n. Every convergent sequence is bounded. Every monotonic and bounded sequence has a limit. Every convergent sequence has a unique limit.

Example 7 . Subsequence
is increasing and limited. She has a limit
=e.

Number e called Euler number and approximately equal to 2.718 28.

Test 9 . The sequence 1, 4, 9, 16,  is:

1) convergent;

2) divergent;

3) limited;

Test 10 . Subsequence
is:

1) convergent;

2) divergent;

3) limited;

4) arithmetic progression;

5) geometric progression.

Test 11 . Subsequence is not:

1) convergent;

2) divergent;

3) limited;

4) harmonic.

Test 12 . Limit of a sequence given by a common term
equal.