Next sequence definition. Defining a number sequence

Subsequence

noun, and., used compare often

Morphology: (no) what? sequences, what? sequences, (see) what? subsequence, how? sequence, about what? about sequence; pl. What? sequences, (no) what? sequences, what? sequences, (see) what? sequences, how? sequences, about what? about sequences

1. Consistency called a row in which one element is located next to another.

Continuous sequence. | Chronological sequence. | Remember the sequence of events. | Consistency in reasoning. | Consistency in actions.

2. In mathematics, computer science sequence name a series of numbers, information elements of a certain type.

Infinite number sequence. | Consistency limit. | A structure is an object consisting of a sequence of named members, each member can be of any type.

Explanatory dictionary of the Russian language by Dmitriev. D. V. Dmitriev. 2003.

Synonyms:

See what “sequence” is in other dictionaries:

A sequence is a set of elements of a certain set: for each natural number you can specify an element of this set; this number is the number of the element and indicates the position of this element in the sequence; for anyone... ... Wikipedia

SUBSEQUENCE. In I.V. Kireevsky’s article “The Nineteenth Century” (1830) we read: “From the very fall of the Roman Empire to our times, the enlightenment of Europe appears to us in gradual development and in uninterrupted sequence” (vol. 1, p.... ... History of words

SEQUENCE, sequences, plural. no, female (book). distracted noun to sequential. A sequence of events. Consistency in the changing tides. Consistency in reasoning. Ushakov's Explanatory Dictionary.... ... Ushakov's Explanatory Dictionary

Constancy, continuity, logic; row, progression, conclusion, series, string, turn, chain, chain, cascade, relay race; persistence, validity, set, methodicality, arrangement, harmony, tenacity, subsequence, connection, queue,... ... Synonym dictionary

SEQUENCE, numbers or elements arranged in an organized manner. Sequences can be finite (having a limited number of elements) or infinite, such as the complete sequence of natural numbers 1, 2, 3, 4 ....... ... Scientific and technical encyclopedic dictionary

SEQUENCE, a set of numbers (mathematical expressions, etc.; they say: elements of any nature), numbered by natural numbers. The sequence is written as x1, x2,..., xn,... or briefly (xi) ... Modern encyclopedia

One of the basic concepts of mathematics. The sequence is formed by elements of any nature, numbered with natural numbers 1, 2, ..., n, ..., and written as x1, x2, ..., xn, ... or briefly (xn) ... Big Encyclopedic Dictionary

Subsequence- SEQUENCE, a set of numbers (mathematical expressions, etc.; they say: elements of any nature), numbered with natural numbers. The sequence is written as x1, x2, ..., xn, ... or briefly (xi). ... Illustrated Encyclopedic Dictionary

SEQUENCE, and, female. 1. See sequential. 2. In mathematics: an infinite ordered set of numbers. Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary

English succession/sequence; German Konsequenz. 1. The order of one after another. 2. One of the basic concepts of mathematics. 3. The quality of correct logical thinking, in which reasoning is free from internal contradictions in one and the other... ... Encyclopedia of Sociology

Subsequence- “a function defined on the set of natural numbers, the set of values of which can consist of elements of any nature: numbers, points, functions, vectors, sets, random variables, etc., numbered by natural numbers... Economic and mathematical dictionary

Books

We build a sequence. Kittens. 2-3 years. Game "Kittens". We build a sequence. Level 1. Series "Preschool education". Cheerful kittens decided to sunbathe on the beach! But they can’t divide the places. Help them...

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:

Test 7 .
is:

Test 8 . Common member of the sequence
is:

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.

Material from Wikipedia - the free encyclopedia

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 automata.

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

Definition

Let some set be given $X$ elements of arbitrary nature. | Any mapping $f\colon\mathbb(N)\to X$ set of natural numbers $\mathbb(N)$ to a given set $X$ called sequence(elements of the set $X$ ).

Image of a natural number $n$ , namely, the element $x_n=f(n)$ , called $n$ -th member or sequence element, and the ordinal number of a member of the sequence is its index.

Related definitions

Subset $f\left[\mathbb(N)\right]$ sets $X$ , which is formed by the elements of the sequence, is called sequence carrier: while the index runs through the set of natural numbers, the point “representing” the sequence “moves” along the carrier.

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

Do not mix the sequence carrier and the sequence itself! For example, dot $a\in X$ as a one-point subset $$a$\subset X$ is the carrier of a stationary sequence of the form $a,a,a,\dots$ .

Any set mapping $\mathbb(N)$ into itself is also a sequence.

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

Designations

Sequences of the form

$x_1,\quad x_2,\quad x_3,\quad\dots$

It is customary to write compactly using parentheses:

$(x_n)$ or $(x_n)_(n=1)^(\infty)$

Curly braces are sometimes used:

$$x_n$_(n=1)^(\infty)$

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

$(x_n)_(n=1)^N$ ,

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

Write a review about the article "Sequence"

Notes

Literature

Sequence // Encyclopedic Dictionary of Young Mathematicians / Comp. A. P. Savin. - M.: Pedagogy, 1985. - P. 242-245. - 352 s.

Sequences and series

Sequences

Rows, main

Number series (actions with number series)

Functional series

Other types of rows

Signs of convergence

Passage characterizing the Sequence

Among the people being selected for the subject of conversation, Julie's company ended up with the Rostovs.
“They say their affairs are very bad,” said Julie. - And he is so stupid - the count himself. The Razumovskys wanted to buy his house and his property near Moscow, and all this drags on. He is treasured.
“No, it seems that the sale will take place one of these days,” someone said. – Although now it’s crazy to buy anything in Moscow.
- From what? – said Julie. – Do you really think that there is a danger for Moscow?
- Why are you going?
- I? That's strange. I’m going because... well, because everyone is going, and then I’m not Joan of Arc or an Amazon.
- Well, yes, yes, give me some more rags.
“If he manages to get things done, he can pay off all his debts,” the militiaman continued about Rostov.
- A good old man, but very pauvre sire [bad]. And why do they live here for so long? They had long wanted to go to the village. Does Natalie seem to be well now? – Julie asked Pierre, smiling slyly.
“They are expecting a younger son,” said Pierre. “He joined Obolensky’s Cossacks and went to Bila Tserkva. A regiment is being formed there. And now they transferred him to my regiment and are waiting for him every day. The Count has long wanted to go, but the Countess will never agree to leave Moscow until her son arrives.
“I saw them the other day at the Arkharovs’. Natalie looked prettier and cheerful again. She sang one romance. How easy it is for some people!
-What's going on? – Pierre asked displeasedly. Julie smiled.
“You know, Count, that knights like you only exist in the novels of Madame Suza.”
- Which knight? From what? – Pierre asked, blushing.
- Well, come on, dear Count, c "est la fable de tout Moscou. Je vous admire, ma parole d" honneur. [all of Moscow knows this. Really, I'm surprised at you.]
- Fine! Fine! - said the militiaman.
- OK then. You can't tell me how boring it is!
“Qu"est ce qui est la fable de tout Moscou? [What does all of Moscow know?] - Pierre said angrily, getting up.
- Come on, Count. You know!
“I don’t know anything,” said Pierre.
– I know that you were friends with Natalie, and that’s why... No, I’m always friendlier with Vera. Cette chere Vera! [This sweet Vera!]
“Non, madame,” Pierre continued in a dissatisfied tone. “I didn’t take on the role of Rostova’s knight at all, and I haven’t been with them for almost a month.” But I don't understand cruelty...
“Qui s"excuse - s"accuse, [Whoever apologizes, blames himself.] - Julie said, smiling and waving the lint, and so that she had the last word, she immediately changed the conversation. “What, I found out today: poor Marie Volkonskaya arrived in Moscow yesterday. Did you hear she lost her father?
- Really! Where is she? “I would very much like to see her,” said Pierre.
– I spent the evening with her yesterday. Today or tomorrow morning she is going to the Moscow region with her nephew.
- Well, how is she? - said Pierre.
- Nothing, I’m sad. But do you know who saved her? This is a whole novel. Nicholas Rostov. They surrounded her, wanted to kill her, wounded her people. He rushed in and saved her...
“Another novel,” said the militiaman. “This general elopement was decidedly done so that all the old brides would get married.” Catiche is one, Princess Bolkonskaya is another.
“You know that I really think that she is un petit peu amoureuse du jeune homme.” [a little bit in love with a young man.]
- Fine! Fine! Fine!
– But how can you say this in Russian?..

When Pierre returned home, he was given two Rastopchin posters that had been brought that day.
The first said that the rumor that Count Rostopchin was prohibited from leaving Moscow was unfair and that, on the contrary, Count Rostopchin was glad that ladies and merchant wives were leaving Moscow. “Less fear, less news,” the poster said, “but I answer with my life that there will be no villain in Moscow.” These words clearly showed Pierre for the first time that the French would be in Moscow. The second poster said that our main apartment was in Vyazma, that Count Wittschstein defeated the French, but that since many residents want to arm themselves, there are weapons prepared for them in the arsenal: sabers, pistols, guns, which residents can get at a cheap price. The tone of the posters was no longer as playful as in Chigirin’s previous conversations. Pierre thought about these posters. Obviously, that terrible thundercloud, which he called upon with all the strength of his soul and which at the same time aroused involuntary horror in him - obviously this cloud was approaching.

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.

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.

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 unlimitedly in absolute value, but do not have a 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, even-numbered terms 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 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 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 stated on the page

Next sequence definition. Defining a number sequence

See what “sequence” is in other dictionaries:

Books

Number sequence limit

Definition

Related definitions

Comments

Designations

see also

Write a review about the article "Sequence"

Notes

Literature

Passage characterizing the Sequence

Definition

Related definitions

Comments

Designations

see also

See what “Sequence” is in other dictionaries:

Books

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