What number sequences are called unlimited. Number sequences. See what “Sequence” is in other dictionaries

Lecture 8. Numerical sequences.

Definition8.1. If each value is associated according to a certain law with a certain real numberx n , then the set of numbered real numbers

– abbreviation
, (8.1)

we'll callnumerical sequence or just a sequence.

Single numbers x n  elements or members of a sequence (8.1).

The sequence can be given by a common term formula, for example:
or
. The sequence can be specified ambiguously, for example, the sequence –1, 1, –1, 1, ... can be specified by the formula
or
. Sometimes a recurrent method of specifying a sequence is used: the first few terms of the sequence and a formula for calculating the next elements are specified. For example, a sequence defined by the first element and the recurrence relation
(arithmetic progression). Consider a sequence called near Fibonacci: the first two elements are set x 1 =1, x 2 =1 and recurrence relation
at any
. We get a sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, …. For such a series it is quite difficult to find a formula for the common term.

8.1. Arithmetic operations with sequences.

Consider two sequences:

(8.1)

Definition 8.2. Let's callproduct of the sequence
per number msubsequence
. Let's write it like this:
.

Let's call the sequence sum of sequences (8.1) and (8.2), we write it like this: ; similarly
let's call sequence difference (8.1) and (8.2);
 product of sequences (8.1) and (8.2);  private sequences (8.1) and (8.2) (all elements
).

8.2. Bounded and unbounded sequences.

The set of all elements of an arbitrary sequence
forms a certain number set that can be bounded from above (from below) and for which definitions similar to those introduced for real numbers are valid.

Definition 8.3. Subsequence
calledbounded above , If ; M  top edge.

Definition 8.4. Subsequence
calledbounded below , If ;m  bottom edge.

Definition 8.5.Subsequence
calledlimited , if it is bounded both above and below, that is, if there are two real numbers M andm such that each element of the sequence
satisfies the inequalities:

, (8.3)

mAndM– bottom and top edges
.

Inequalities (8.3) are called condition of boundedness of the sequence
.

For example, the sequence
limited and
unlimited.

♦ Statement 8.1.
is limited .

Proof. Let's choose
. According to Definition 8.5, the sequence
will be limited. ■

Definition 8.6. Subsequence
calledunlimited , if for any positive (no matter how large) real number A there is at least one element of the sequencex n , satisfying the inequality:
.

For example, the sequence 1, 2, 1, 4, …, 1, 2 n, … unlimited, because limited only from below.

8.3. Infinitely large and infinitesimal sequences.

Definition 8.7. Subsequence
calledinfinitely large , if for any (no matter how large) real number A there is a number
such that in front of everyone
elementsx n
.

☼ Remark 8.1. If a sequence is infinitely large, then it is unbounded. But one should not think that any unbounded sequence is infinitely large. For example, the sequence
is not limited, but is not infinitely large, because condition
does not hold for all even n. ☼

 Example 8.1.
is infinitely large. Let's take any number A>0. From inequality
we get n>A. If you take
, then for everyone n>N inequality will be satisfied
, that is, according to Definition 8.7, the sequence
infinitely large. 

Definition 8.8. Subsequence
calledinfinitesimal , if for
(as small as you like ) there is a number
such that in front of everyone
elements this sequence satisfies the inequality
.

 Example 8.2. Let us prove that the sequence infinitely small.

Let's take any number
. From inequality
we get . If you take
, then for everyone n>N inequality will be satisfied
. 

♦ Statement 8.2. Subsequence
is infinitely large at
and infinitesimal at
.

Proof.

1) Let first
:
, Where
. According to Bernoulli's formula (example 6.3, paragraph 6.1.)
. Fix an arbitrary positive number A and select a number from it N such that the following inequality is true:

,
,
,
.

Because
, then by the property of the product of real numbers for all

.

Thus, for
there is such a number
that in front of everyone

– infinitely large at
.

2) Consider the case
,
(at q=0 we have a trivial case).

Let
, Where
, according to Bernoulli's formula
or
.

We fix
,
and choose
such that

,
,
.

For

. Let's indicate this number N that in front of everyone

, that is, when
subsequence
infinitely small. ■

8.4. Basic properties of infinitesimal sequences.

♦ Theorem 8.1.Sum

And

Proof. We fix ;
– infinitesimal

,

– infinitesimal

. Let's choose
. Then at

,
,
. ■

♦ Theorem 8.2. Difference
two infinitesimal sequences
And
there is an infinitesimal sequence.

For proof theorem, it is enough to use the inequality . ■

Consequence.The algebraic sum of any finite number of infinitesimal sequences is an infinitesimal sequence.

♦ Theorem 8.3.The product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence.

Proof.
– limited,
is an infinitesimal sequence. We fix ;
,
;
: at
fair
. Then
. ■

♦ Theorem 8.4.Every infinitesimal sequence is bounded.

Proof. We fix Let some number . Then
for all numbers n, which means the sequence is limited. ■

Consequence. The product of two (and any finite number) of infinitesimal sequences is an infinitesimal sequence.

♦ Theorem 8.5.

If all the elements of an infinitesimal sequence
equal to the same numberc, then c= 0.

Proof The theorem is carried out by contradiction, if we denote
. ■

♦ Theorem 8.6. 1) If
is an infinitely large sequence, then, starting from some numbern, the quotient is defined two sequences
And
, which is an infinitesimal sequence.

2) If all the elements of an infinitesimal sequence
are different from zero, then the quotient two sequences
And
is an infinitely large sequence.

Proof.

1) Let
– an infinitely large sequence. We fix ;
or
at
. Thus, by definition 8.8 the sequence – infinitesimal.

2) Let
is an infinitesimal sequence. Let's assume that all elements
are different from zero. We fix A;
or
at
. By definition 8.7 the sequence infinitely large. ■

Subsequence

Subsequence- This kit elements of some set:

for each natural number you can specify an element of a given set;
this number is the number of the element and indicates the position of this element in the sequence;
For any element (member) of a sequence, you can specify the next element of the sequence.

So the sequence turns out to be the result consistent selection of elements of a given set. And, if any set of elements is finite, and we talk about a sample of finite volume, then the sequence turns out to be a sample of infinite volume.

A sequence is by its nature a mapping, so it should not be confused with a set that “runs through” the sequence.

In mathematics, many different sequences are considered:

time series of both numerical and non-numerical nature;
sequences of elements of metric space
sequences of functional space elements
sequences of states of control systems and machines.

The purpose of studying all possible sequences is to search for patterns, predict future states and generate sequences.

Definition

Let a certain set of elements of arbitrary nature be given. | Any mapping from a set of natural numbers to a given set is called sequence(elements of the set).

The image of a natural number, namely, the element, is called - th member or sequence element, and the ordinal number of a member of the sequence is its index.

Related definitions

If we take an increasing sequence of natural numbers, then it can be considered as a sequence of indices of some sequence: if we take the elements of the original sequence with the corresponding indices (taken from the increasing sequence of natural numbers), then we can again get a sequence called subsequence given sequence.

Comments

In mathematical analysis, an important concept is the limit of a number sequence.

Designations

Sequences of the form

It is customary to write compactly using parentheses:

Curly braces are sometimes used:

Allowing some freedom of speech, we can also consider finite sequences of the form

which represent the image of the initial segment of a sequence of natural numbers.

Definition

Number sequence (xn)- this is a law (rule), according to which, for every natural number n = 1, 2, 3, . . . a certain number x n is assigned.
The element x n is called nth term or an element of a sequence.

The sequence is denoted as the nth term enclosed in curly braces: . The following designations are also possible: . They explicitly indicate that the index n belongs to the set of natural numbers and the sequence itself has an infinite number of terms. Here are some example sequences:
, , .

In other words, a number sequence is a function whose domain of definition is the set of natural numbers. The number of elements of the sequence is infinite. Among the elements there may also be members that have the same meanings. Also, a 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 of how sequences behave when n tends to infinity: . 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 a sequence with a common term. Here are its first few members:
.
As the number n increases, the elements of this sequence increase indefinitely in absolute value, but don't have constant sign. That is, this sequence tends to: at .

Examples of sequences converging to a finite number

Consider the sequence. Her common member. The first terms have the following form:
.
It can be seen that as the number n increases, the elements of this sequence approach their limiting value a = 0 : at . So each subsequent term is closer to zero than the previous one. In a sense, we can consider that there is an approximate value for the number a = 0 with error. It is clear that as n increases, this error tends to zero, that is, by choosing n, the error can be made as small as desired. Moreover, for any given error ε > 0 you can specify a number N such 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. Her common member. Here are some of its first members:
.
In this sequence, terms with even numbers are equal to zero. Terms with odd n are equal. Therefore, as n increases, their values approach the limiting value a = 0 . This also follows from the fact that
.
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 = 0 by an amount 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 terms with even numbers:
,
converge to the value a 1 = 0 . Odd-numbered members:
,
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 let's look at a more interesting sequence. Let's take a segment on the number line. Let's divide it in half. We get two segments. Let
.
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 (0; 1) . 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.

Then from the original sequence one can select a subsequence that will converge to an arbitrary point from the interval . 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 = 0 you can choose the following subsequence:
.
= 0 .

For point a = 1 Let's choose the following subsequence:
.
The terms 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 appear 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 associate each natural number n with a pair of numbers p and q so that any pair p and q is included in our sequence.

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 c will correspond to a rational number. 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 (0; 0) (see picture). In this case, the lower parts of the squares with q < 1 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 the top part of the next square:

.
We number the top part of the following square:

.
And so on.

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 - 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 that converge to different numbers, 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. The exact definition of convergence is discussed on the page Defining the Limit of a Sequence. Related properties and theorems are outlined on the page Limit of a sequence - basic theorems and properties.

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:

All limits are abbreviated as lim.
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:

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.
If a series of numbers has a limit, then the sequence of these numbers is limited.
The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
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:

The sum of any number of any number of small quantities will also be a small quantity.
The sum of any number of large quantities will be an infinitely large quantity.
The product of arbitrarily small quantities is infinitesimal.
The product of any number of large numbers is infinitely large.
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.

What number sequences are called unlimited. Number sequences. See what “Sequence” is in other dictionaries

Definition

Related definitions

Comments

Designations

see also

See what “Sequence” is in other dictionaries:

Books

Definition

Sequence Examples

Examples of infinitely increasing sequences

Examples of sequences converging to a finite number

Examples of divergent sequences

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

Sequence containing all rational numbers

Conclusion

What are sequences and where is their limit?

How is the number sequence constructed?

Arithmetic progression as part of sequences

Types of sequences

Determining the Sequence Limit

General designation for the limit of sequences

Uncertainty and certainty of the limit

What is a neighborhood?

Theorems

Proof of sequences

Or maybe he's not there?

Monotonic sequence

Limit of a convergent and bounded sequence

Limit of a monotonic sequence

Various actions with limits

Properties of sequence quantities