The definition of a numerical sequence is given. Examples of infinitely increasing, convergent, and divergent sequences are considered. A sequence containing all rational numbers is considered.

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See also:

Definition

Numeric sequence ( x n )- this is the law (rule), according to which, for each natural number n = 1, 2, 3, . . . some number x n is assigned.
The element x n is called the nth member or element of the sequence.

The sequence is denoted as the nth member enclosed in curly brackets: . The following designations are also possible: . They explicitly state that the index n belongs to the set of natural numbers and that the sequence itself has an infinite number of members. Here are some examples of sequences:
, , .

In other words, a numerical sequence is a function whose domain is the set of natural numbers. The number of elements in the sequence is infinite. Among the elements, there may also be members that have the same value. Also, the sequence can be considered as a numbered set of numbers, consisting of an infinite number of members.

We will be mainly interested in the question - how sequences behave when n tends to infinity: . This material is presented in the Sequence limit - basic theorems and properties section. And here we will look at some examples of sequences.

Sequence examples

Examples of infinitely increasing sequences

Let's consider a sequence. The general term of this sequence is . Let's write out the first few terms:
.
It can be seen that as the number n grows, the elements increase indefinitely towards positive values. We can say that this sequence tends to : at .

Now consider a sequence with a common term . Here are some of its first members:
.
As the number n grows, the elements of this sequence increase in absolute value indefinitely, but do not have a constant sign. That is, this sequence tends to : at .

Examples of sequences converging to a finite number

Let's consider a sequence. Its common member The first terms are as follows:
.
It can be seen that as the number n grows, the elements of this sequence approach their limit value a = 0 : at . So each subsequent term is closer to zero than the previous one. In a sense, we can assume that there is an approximate value for the number a = 0 with an error. It is clear that as n grows, this error tends to zero, that is, by choosing n, the error can be made arbitrarily small. Moreover, for any given error ε > 0 it is possible to specify such a number N , that for all elements with numbers greater than N : , the deviation of the number from the limit value a will not exceed the error ε : .

Next, consider the sequence. Its common member Here are some of its first members:
.
In this sequence, even-numbered terms are zero. Members with odd n are . Therefore, as n grows, their values ​​approach the limiting value a = 0 . This also follows from the fact that
.
As in the previous example, we can specify an arbitrarily small error ε > 0 , for which it is possible to find such a number N that elements with numbers greater than N will deviate from the limit value a = 0 by a value not exceeding the specified error. Therefore, this sequence converges to the value a = 0 : at .

Examples of divergent sequences

Consider a sequence with the following common term:

Here are its first members:


.
It can be seen that the terms with even numbers:
,
converge to the value a 1 = 0 . Members with odd numbers:
,
converge to the value a 2 = 2 . The sequence itself, as n grows, does not converge to any value.

Sequence with terms distributed in the interval (0;1)

Now consider a more interesting sequence. Take a segment on the number line. Let's split it in half. We get two segments. Let be
.
Each of the segments is again divided in half. We get four segments. Let be
.
Divide each segment in half again. Let's take


.
Etc.

As a result, we obtain a sequence whose elements are distributed in an open interval (0; 1) . Whatever point we take from the closed interval , we can always find members of the sequence that are arbitrarily close to this point, or coincide with it.

Then from the original sequence one can single out a subsequence that will converge to an arbitrary point from the interval . That is, as the number n grows, the members of the subsequence will come closer and closer to the preselected point.

For example, for point a = 0 you can choose the following subsequence:
.
= 0 .

For point a = 1 choose the following subsequence:
.
The members of this subsequence converge to the value a = 1 .

Since there are subsequences that converge to different values, the original sequence itself does not converge to any number.

Sequence containing all rational numbers

Now let's construct a sequence that contains all rational numbers. Moreover, each rational number will be included in such a sequence an infinite number of times.

The rational number r can be represented as follows:
,
where is an integer; - natural.
We need to assign to each natural number n a pair of numbers p and q so that any pair of p and q is included in our sequence.

To do this, draw axes p and q on the plane. We draw grid lines through integer values ​​p and q . Then each node of this grid with will correspond to a rational number. The whole 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 a single node. This is easy to do if we number the nodes according to the squares whose centers are located at the point (0; 0) (see picture). In this case, the lower parts of the squares with q < 1 we don't need. Therefore, they are not shown in the figure.

So, for the upper side of the first square we have:
.
Next, we number the upper part of the next square:

.
We number the upper part of the next square:

.
Etc.

In this way we get a sequence containing all rational numbers. It can be seen 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 elements of which are equal to a predetermined rational number. Since the sequence we have constructed has subsequences converging to different numbers, the sequence does not converge to any number.

Conclusion

Here we have given a precise definition of the numerical sequence. We also touched upon the issue of its convergence, based on intuitive ideas. The exact definition of convergence is discussed on the page Determining the Limit of a Sequence. Related properties and theorems are outlined on the Sequence Limit - Basic Theorems and Properties page.

See also:

Let be X (\displaystyle X) is either the set of real numbers R (\displaystyle \mathbb (R) ), or the set of complex numbers C (\displaystyle \mathbb (C) ). Then the sequence ( x n ) n = 1 ∞ (\displaystyle \(x_(n)\)_(n=1)^(\infty )) set elements X (\displaystyle X) called numerical sequence.

Examples

Operations on sequences

Subsequences

Subsequence sequences (x n) (\displaystyle (x_(n))) is the sequence (x n k) (\displaystyle (x_(n_(k)))), where (n k) (\displaystyle (n_(k))) is an increasing sequence of elements of the set of natural numbers.

In other words, a subsequence is obtained from a sequence by removing a finite or countable number of elements.

Examples

• The sequence of prime numbers is a subsequence of the sequence of natural numbers.
• The sequence of natural numbers that are multiples of is a subsequence of the sequence of even natural numbers.

Properties

Sequence limit point is a point in any neighborhood of which there are infinitely many elements of this sequence. For convergent numerical sequences, the limit point coincides with the limit.

Sequence limit

Sequence limit is the object that the members of the sequence approach as the number increases. Thus, in an arbitrary topological space, the limit of a sequence is an element in any neighborhood of which all members of the sequence lie, starting from some one. In particular, for numerical sequences, the limit is a number in any neighborhood of which all members of the sequence lie, starting from some one.

Fundamental sequences

Fundamental sequence (self-convergent sequence , Cauchy sequence ) is a sequence of elements of a metric space , in which, for any predetermined distance, there is such an element, the distance from which to any of the elements following it does not exceed the given one. For numerical sequences, the concepts of fundamental and convergent sequences are equivalent, but in the general case this is not the case.

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 the middle school, letter designations come into play, and in the older one they can no longer be dispensed with.

But today we will talk about what all known mathematics is based on. About the 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 such a construction 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 to the store, this is one sequence. And if one person suddenly leaves this queue, then this is a different queue, a different order.

The word "limit" is also easily interpreted - this is the end of something. However, in mathematics, the limits of sequences are those values ​​on the number line that a sequence of numbers tends to. Why strives and does 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, it is a series of members of some set.

How is a number sequence built?

The simplest example of a number sequence might look like this: 1, 2, 3, 4, …n…

In most cases, for practical purposes, sequences are built from numbers, and each next member of the series, let's denote it by X, has its own name. For example:

x 1 - the first member of the sequence;

x 2 - the second member of the sequence;

x 3 - the third member;

x n is the nth member.

In practical methods, the sequence is given by a general formula in which there is some variable. For example:

X n \u003d 3n, then the series of numbers itself will look like this:

It is worth remembering that in the general notation of sequences, you can use any Latin letters, and 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 delve deeper into the very concept of such a number series, which everyone encountered when they were in the middle classes. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.

Task: “Let a 1 \u003d 15, and the step of the progression of the number series d \u003d 4. Build the first 4 members of this row"

Solution: a 1 = 15 (by condition) is the first member of the progression (number series).

and 2 = 15+4=19 is the second member of the progression.

and 3 \u003d 19 + 4 \u003d 23 is the third term.

and 4 \u003d 23 + 4 \u003d 27 is the fourth term.

However, with 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 \u003d a 1 + d (n-1). In this case, a 125 \u003d 15 + 4 (125-1) \u003d 511.

Sequence types

Most of the sequences are endless, it's worth remembering for a lifetime. There are two interesting types of number series. The first is given by the formula a n =(-1) n . Mathematicians often refer to this flasher sequences. Why? Let's check its numbers.

1, 1, -1 , 1, -1, 1, etc. With this example, it becomes clear that numbers in sequences can easily be repeated.

factorial sequence. It is easy to guess that there is a factorial in the formula that defines the sequence. For example: and n = (n+1)!

Then the sequence will look like this:

and 2 \u003d 1x2x3 \u003d 6;

and 3 \u003d 1x2x3x4 \u003d 24, etc.

A sequence given by an arithmetic progression is called infinitely decreasing if the inequality -1 is observed for all its members

and 3 \u003d - 1/8, etc.

There is even a sequence consisting of the same number. So, and n \u003d 6 consists of an infinite number of sixes.

Determining the Limit of a Sequence

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, consider the limit for a linear function in detail:

1. All limits are abbreviated as lim.
2. The limit entry consists of the abbreviation lim, some 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: it is a certain number, to which all members of the sequence infinitely approach. Simple example: and 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 that its limit is equal to infinity as x→∞, and this should be written as follows:

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 more and more close to one (0.1, 0.2, 0.9, 0.986). It can be seen from this series 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 tasks.

General notation for the limit of sequences

Having analyzed the limit of the numerical sequence, its definition and examples, we 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 existence 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 consolidate the material, read the formula aloud.

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 x values ​​(increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. It turns out a rather strange fraction:

But is it really so? Calculating the limit of the numerical sequence in this case seems easy enough. It would be possible to leave everything as it is, because the answer is ready, and it was received on reasonable terms, 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.

Divide both the numerator and the denominator by the variable to the highest degree. In this case, we divide the fraction by x 1.

Next, let's find what value each term containing the variable tends to. In this case, fractions are considered. As x→∞, the value of each of the fractions tends to zero. When making a paper in writing, it is worth making the following footnotes:

The following expression is obtained:

Of course, the fractions containing x did not become zeros! But their value is so small that it is quite permissible not to take it into account in the calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero.

What is a neighborhood?

Let us assume that the professor has at his disposal a complex sequence, given, obviously, by a no less complex formula. The professor found the answer, but does it fit? After all, all people make mistakes.

Auguste Cauchy came up with a great way to prove the limits of sequences. His method was called neighborhood operation.

Suppose that there is some point a, its neighborhood in both directions on the real line is equal to ε ("epsilon"). Since the last variable is distance, its value is always positive.

Now let's set some sequence x n and suppose that the tenth member of the sequence (x 10) is included in the neighborhood of a. How to write this fact in mathematical language?

Suppose 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 is time to explain in practice the formula mentioned above. It is fair to call some number a the end point of a sequence if the inequality ε>0 holds for any of its limits, 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 limits of a sequence, to prove or disprove a ready 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, you can significantly facilitate the process of solving or proving:

1. Uniqueness of the limit of a sequence. Any sequence can have only one limit or not 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 quotient limit of two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.

Sequence Proof

Sometimes it is required 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 equal to zero.

According to the above rule, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим:

Let's express n in terms of "epsilon" to show the existence of a certain number and prove the existence of a sequence limit.

At this stage, it is important to recall that "epsilon" and "en" are positive numbers and not equal to zero. Now you can continue further transformations using the knowledge about inequalities gained in high school.

Whence it turns 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 proved 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 this we can safely assert that the number a is the limit of the given sequence. Q.E.D.

With such a convenient method, you can prove the limit of a numerical sequence, no matter how complicated it may seem at first glance. The main thing is not to panic at the sight of the task.

Or maybe he doesn't exist?

The existence of a sequence limit is not necessary in practice. It is easy to find such series of numbers that really have no end. For example, the same flasher x n = (-1) n . it is obvious that a sequence consisting of only two digits cyclically repeating cannot have a limit.

The same story is repeated with sequences consisting of a single number, fractional, having in the course of calculations an uncertainty of any order (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculation also takes place. Sometimes rechecking your own solution will help you find the limit of successions.

monotonic sequence

Above, we considered several examples of sequences, methods for solving them, and now let's try to take a more specific case and call it a "monotone sequence".

Definition: it is fair to call any sequence monotonically increasing if it satisfies the strict inequality x n< 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 is easier to understand this with examples.

The sequence given by the formula x n \u003d 2 + n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.

And if we take x n \u003d 1 / n, then we get a series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing sequence.

Limit of 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 quantity (real or complex). If you draw a sequence diagram, then at a certain point it will, as it were, converge, tend to turn into a certain value. Hence the name - convergent sequence.

Monotonic sequence limit

Such a sequence may or may not have a limit. First, it is useful to understand when it is, from here you can start when proving the absence of a limit.

Among monotonic sequences, convergent and divergent are distinguished. Convergent - this is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent - a sequence that has no limit in its set (neither real nor complex).

Moreover, the sequence converges if its upper and lower limits converge in a geometric representation.

The limit of a convergent sequence can in many cases be equal to zero, since any infinitesimal sequence has a known limit (zero).

Whichever convergent sequence you take, they are all bounded, but far from all bounded sequences converge.

The sum, difference, product of two convergent sequences is also a convergent sequence. However, the quotient can also converge if it is defined!

Various actions with limits

Sequence limits are as significant (in most cases) as numbers and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits.

First, just 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 is true: 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 equal to zero. After all, if the limit of sequences is equal to zero, then division by zero will turn out, which is impossible.

Sequence Value Properties

It would seem that the limit of the numerical sequence has already been analyzed in some detail, but such phrases 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 infinitely small, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such values ​​have their own characteristics. The properties of the limit of a sequence having arbitrary small or large values ​​are as follows:

1. The sum of any number of arbitrarily small quantities will also be a small quantity.
2. The sum of any number of large values ​​will be an infinitely large value.
3. The product of arbitrarily small quantities is infinitely small.
4. The product of arbitrarily large numbers is an infinitely large quantity.
5. If the original sequence tends to an infinite number, then the reciprocal of it 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 sequences are a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution of such expressions. Starting small, over time, you can reach big heights.

Numerical sequence is called a numerical function defined on the set of natural numbers .

If the function is given on the set of natural numbers
, then the set of values ​​of the function will be countable and each number
number is matched
. In this case, we say that given numerical sequence. Numbers are called elements or members of a sequence, and the number - general or -th member of the sequence. Each element has a follower
. This explains the use of the term "sequence".

The sequence is usually specified either by listing its elements, or by indicating the law by which the element with the number is calculated , i.e. indicating the formula th member .

Example.Subsequence
can be given by the formula:
.

Usually sequences are denoted as follows: etc., where the formula of its th member.

Example.Subsequence
‑this is the sequence

The set of all elements of a sequence
denoted
.

Let be
and
- two sequences.

With ummah sequences
and
call the sequence
, where
, i.e..

R aznosti of these sequences is called the sequence
, where
, i.e..

If a and ‑ constants, then the sequence
,
called linear combination sequences
and
, i.e.

work sequences
and
call the sequence -th member
, i.e.
.

If a
, then it is possible to determine private
.

Sum, difference, product and quotient of sequences
and
they are called algebraiccompositions.

Example.Consider the sequences
and
, where. Then
, i.e. subsequence
has all elements equal to zero.

,
, i.e. all elements of the product and the quotient are equal
.

If we cross out some elements of the sequence
so that there are an infinite number of elements left, then we get another sequence, called subsequence sequences
. If we cross out the first few elements of the sequence
, then the new sequence is called remainder.

Subsequence
limitedabove(from below) if the set
limited from above (from below). The sequence is called limited if it is bounded above and below. A sequence is bounded if and only if any of its remainder is bounded.

Converging Sequences

They say that subsequence
converges if there is a number such that for any
there is such
, which for any
, the following inequality holds:
.

Number called sequence limit
. At the same time, they record
or
.

Example.
.

Let us show that
. Set any number
. Inequality
performed for
, such that
that the definition of convergence holds for the number
. Means,
.

In other words
means that all members of the sequence
with sufficiently large numbers differs little from the number , i.e. starting from some number
(when) the elements of the sequence are in the interval
, which is called -neighborhood of the point .

Subsequence
, whose limit is equal to zero (
, or
at
) is called infinitesimal.

As applied to infinitesimals, the following statements are true:

The sum of two infinitesimals is infinitesimal;

The product of an infinitesimal by a bounded value is an infinitesimal.

Theorem .In order for the sequence
had a limit, it is necessary and sufficient that
, where - constant; - infinitely small .

Main properties of convergent sequences:

Properties 3. and 4. generalize to the case of any number of convergent sequences.

Note that when calculating the limit of a fraction whose numerator and denominator are linear combinations of powers , the limit of the fraction is equal to the limit of the ratio of the highest terms (i.e., the terms containing the largest powers numerator and denominator).

Subsequence
called:

All such sequences are called monotonous.

Theorem . If the sequence
increases monotonically and is bounded from above, then it converges and its limit is equal to its greatest upper bound; if the sequence is decreasing and bounded below, then it converges to its greatest lower bound.

If a function is defined on the set of natural numbers N, then such a function is called an infinite number sequence. Usually, a numerical sequence is denoted as (Xn), where n belongs to the set of natural numbers N.

The numerical sequence can be given by a formula. For example, Xn=1/(2*n). Thus, we assign to each natural number n some definite element of the sequence (Xn).

If we now successively take n equal to 1,2,3, …., we get the sequence (Xn): ½, ¼, 1/6, …, 1/(2*n), …

Sequence types

The sequence can be limited or unlimited, increasing or decreasing.

The sequence (Xn) calls limited if there are two numbers m and M such that for any n belonging to the set of natural numbers, the equality m<=Xn

Sequence (Xn), not limited, is called an unbounded sequence.

increasing if for all positive integers n the following equality holds: X(n+1) > Xn. In other words, each member of the sequence, starting from the second, must be greater than the previous member.

The sequence (Xn) is called waning, if for all positive integers n the following equality holds X(n+1)< Xn. Иначе говоря, каждый член последовательности, начиная со второго, должен быть меньше предыдущего члена.

Sequence example

Let's check if the sequences 1/n and (n-1)/n are decreasing.

If the sequence is decreasing, then X(n+1)< Xn. Следовательно X(n+1) - Xn < 0.

X(n+1) - Xn = 1/(n+1) - 1/n = -1/(n*(n+1))< 0. Значит последовательность 1/n убывающая.

(n-1)/n:

X(n+1) - Xn =n/(n+1) - (n-1)/n = 1/(n*(n+1)) > 0. So the sequence (n-1)/n is increasing.