How the sequence is determined. Number sequence

Subsequence

Subsequence- this is kit elements of some set:

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

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

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

Many different sequences are considered in mathematics:

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

The goal of studying all kinds of sequences is to find patterns, predict future states, and generate sequences.

Definition

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

The image of a natural number, namely, an element, is called - th member of 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 a certain sequence: if we take the elements of the original sequence with the corresponding indices (taken from an increasing sequence of natural numbers), then we can again obtain a sequence, which is called subsequence a given sequence.

Comments (1)

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 liberty of speech, one can also consider finite sequences of the form

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

Properties of number sequences.

A numerical sequence is a special case of a numerical function, therefore, a number of properties of functions are considered for sequences as well.

Definition . Subsequence ( y n} is called increasing if each of its members (except the first) is larger than the previous one:

y 1 y 2 y 3 y n y n +1

Definition. The sequence ( y n} is called decreasing if each of its members (except for the first) is less than the previous one:

y 1 > y 2 > y 3 > … > y n> y n +1 > … .

Ascending and descending sequences are united by a common term - monotonic sequences.

Example 1. y 1 = 1; y n= n 2 - ascending sequence.

Thus, the following theorem is true (a characteristic property of an arithmetic progression). A numerical sequence is arithmetic if and only if each of its members, except for the first (and the last in the case of a finite sequence), is equal to the arithmetic mean of the preceding and subsequent members.

Example. At what value x number 3 x + 2, 5x- 4 and 11 x+ 12 form a finite arithmetic progression?

According to the characteristic property, the given expressions must satisfy the relation

5x – 4 = ((3x + 2) + (11x + 12))/2.

The solution to this equation gives x= –5,5. With this value x given expressions 3 x + 2, 5x- 4 and 11 x+ 12 take, respectively, values of –14.5, –31,5, –48,5. It - arithmetic progression, its difference is –17.

Geometric progression.

A numerical sequence, all members of which are nonzero and each member of which, starting from the second, is obtained from the previous term by multiplying by the same number q, is called a geometric progression, and the number q- the denominator of a geometric progression.

Thus, geometric progression Is a numerical sequence ( b n) defined recursively by the relations

b 1 = b, b n = b n –1 q (n = 2, 3, 4…).

(b and q - given numbers, b ≠ 0, q ≠ 0).

Example 1.2, 6, 18, 54, ... - increasing geometric progression b = 2, q = 3.

Example 2. 2, –2, 2, –2,… – geometric progression b= 2,q= –1.

Example 3. 8, 8, 8, 8, ... – geometric progression b= 8, q= 1.

A geometric progression is an ascending sequence if b 1 > 0, q> 1, and decreasing if b 1> 0, 0 q

One of the obvious properties of a geometric progression is that if a sequence is a geometric progression, then a sequence of squares, i.e.

b 1 2 , b 2 2 , b 3 2 , …, b n 2, ... is a geometric progression, the first term of which is b 1 2, and the denominator is q 2 .

Formula n- th term of the geometric progression has the form

b n= b 1 q n– 1 .

You can get a formula for the sum of the members of a finite geometric progression.

Let a finite geometric progression be given

b 1 ,b 2 ,b 3 , …, b n

let be S n - the sum of its members, i.e.

S n= b 1 + b 2 + b 3 + … +b n.

It is assumed that q No. 1. To determine S n an artificial trick is applied: some geometric transformations of the expression are performed S n q.

S n q = (b 1 + b 2 + b 3 + … + b n –1 + b n)q = b 2 + b 3 + b 4 + …+ b n+ b n q = S n+ b n q– b 1 .

Thus, S n q= S n +b n q - b 1 and therefore

This is a formula with ummah n members of a geometric progression for the case when q≠ 1.

At q= 1, the formula can be omitted separately, it is obvious that in this case S n= a 1 n.

The geometric progression is named because each term in it, except for the first, is equal to the geometric mean of the previous and subsequent members. Indeed, since

b n = b n- 1 q;

b n = b n + 1 / q,

hence, b n 2= b n– 1 b n + 1 and the following theorem is true (characteristic property of a geometric progression):

a numerical sequence is a geometric progression if and only if the square of each of its members, except for the first (and the last in the case of a finite sequence), is equal to the product of the previous and subsequent members.

Sequence limit.

Let there be a sequence ( c n} = {1/n}. This sequence is called harmonic, since each of its members, starting from the second, is the harmonic mean between the previous and subsequent members. Geometric mean of numbers a and b there is a number

Otherwise, the sequence is called divergent.

Based on this definition, one can, for example, prove the existence of the limit A = 0 harmonic sequence ( c n} = {1/n). Let ε be an arbitrarily small positive number. The difference is considered

Does such a thing exist N that for everyone n ≥ N inequality 1 / N? If we take as N any natural number exceeding 1/ε then for all n ≥ N inequality 1 / n ≤ 1/ N ε, Q.E.D.

It is sometimes very difficult to prove that a sequence has a limit. The most common sequences are well studied and are listed in reference books. There are important theorems that allow us to conclude that a given sequence has a limit (and even calculate it), based on the sequences already studied.

Theorem 1. If a sequence has a limit, then it is bounded.

Theorem 2. If a sequence is monotone and bounded, then it has a limit.

Theorem 3. If the sequence ( a n} has a limit A, then the sequences ( ca n}, {a n+ s) and (| a n|} have limits cA, A +c, |A| respectively (here c- an arbitrary number).

Theorem 4. If sequences ( a n} and ( b n) have limits equal to A and B pa n + qb n) has a limit pA+ qB.

Theorem 5. If sequences ( a n) and ( b n) have limits equal to A and B respectively, then the sequence ( a n b n) has a limit AB.

Theorem 6. If sequences ( a n} and ( b n) have limits equal to A and B respectively, and, moreover, b n ≠ 0 and B ≠ 0, then the sequence ( a n / b n) has a limit A / B.

Anna Chugainova

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

If the function is set on the set of natural numbers
, then the set of values of the function will be countable and each number
matches the number
... In this case, they say that given numerical sequence... The numbers are called elements or members of the sequence, and the number - general or Th member of the sequence. Every element has a follow-up element
... This explains the use of the term "sequence".

A sequence is usually set either by enumerating its elements, or by specifying the law by which the element with the number is calculated , i.e. indicating its formula Th member .

Example.Subsequence
can be given by the formula:
.

Usually the sequences are designated as follows: etc., where the formula is indicated in brackets th member.

Example.Subsequence
‑this is the sequence

Set of all elements of the sequence
denoted
.

Let be
and
- two sequences.

WITH ummah sequences
and
call sequence
, where
, i.e.

R the abundance these sequences are called the sequence
, where
, i.e.

If and ‑ constant, then the sequence
,
are called linear combination sequences
and
, i.e.

By product sequences
and
call a sequence with -th member
, i.e.
.

If
, then you can define 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 work and the quotient are equal
.

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

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

Converging sequences

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

Number are called limit of sequence
... At the same time, write
or
.

Example.
.

Let us show that
... Let's set any number
... Inequality
performed for
such that
that the definition of convergence is satisfied 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
(for) the elements of the sequence are in the interval
which is called - the neighborhood of the point .

Subsequence
, the limit of which is zero (
, or
at
) is called infinitesimal.

With regard to infinitesimal, the following statements are true:

The sum of two infinitesimal is infinitesimal;

The product of an infinitesimal by a limited quantity is infinitesimal.

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

Basic properties of converging sequences:

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

Note that when calculating the limit of a fraction, the numerator and denominator of which 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 greatest 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 exact upper bound; if the sequence decreases and is bounded from below, then it converges to its exact lower bound.

Introduction ……………………………………………………………………………… 3

1.Theoretical part ……………………………………………………………… .4

Basic concepts and terms ……………………………………………… .... 4

1.1 Types of sequences ……………………………………………… ... 6

1.1.1.Limited and Unlimited Numeric Sequences ... ..6

1.1.2. Monotonicity of sequences ………………………………… 6

1.1.3. Infinitely large and infinitely small sequences …… .7

1.1.4. Properties of infinitesimal sequences ………………… 8

1.1.5. Converging and diverging sequences and their properties ... ... 9

1.2 Limit of sequence ………………………………………………… .11

1.2.1. Sequence limits theorems …………………………………………………………………………………………… 15

1.3. Arithmetic progression ……………………………………………… 17

1.3.1. Properties of the arithmetic progression ………………………………… ..17

1.4 Geometric progression ……………………………………………… ..19

1.4.1. Properties of a geometric progression …………………………………… .19

1.5. Fibonacci numbers ………………………………………………………… ..21

1.5.1 Relationship of Fibonacci numbers with other areas of knowledge …………………… .22

1.5.2. Using a series of Fibonacci numbers to describe animate and inanimate nature …………………………………………………………………………… .23

2. Own research ………………………………………………… .28

Conclusion ……………………………………………………………………… .30

List of used literature ………………………………………… .... 31

Introduction.

Number sequences are a very interesting and educational topic. This topic occurs in quests increased complexity offered to students by authors didactic materials, in problems of mathematical olympiads, entrance exams to Higher Educational establishments and on the exam. I am interested in learning the relationship of mathematical sequences with other areas of knowledge.

Target research work: Expand your knowledge of the number sequence.

1. Consider the sequence;

2. Consider its properties;

3. Consider the analytical task of the sequence;

4. Demonstrate its role in the development of other areas of knowledge.

5. Demonstrate the use of a series of Fibonacci numbers to describe animate and inanimate nature.

1. The theoretical part.

Basic concepts and terms.

Definition. A numerical sequence is a function of the form y = f (x), x О N, where N is a set of natural numbers (or a function of a natural argument), denoted by y = f (n) or y1, y2,…, yn,…. The values y1, y2, y3,… are called, respectively, the first, second, third,… members of the sequence.

The number a is called the limit of the sequence x = (x n) if for an arbitrary predetermined arbitrarily small positive number ε there is a natural number N such that for all n> N the inequality | x n - a |< ε.

If the number a is the limit of the sequence x = (x n), then they say that x n tends to a, and write

A sequence (yn) is called ascending if each of its members (except the first) is larger than the previous one:

y1< y2 < y3 < … < yn < yn+1 < ….

A sequence (yn) is called decreasing if each of its members (except the first) is less than the previous one:

y1> y2> y3>…> yn> yn + 1>….

Ascending and descending sequences are united by a common term - monotonic sequences.

A sequence is called periodic if there exists a natural number T such that, starting from some n, the equality yn = yn + T holds. The number T is called the length of the period.

An arithmetic progression is a sequence (an), each term of which, starting from the second, is equal to the sum of the previous term and the same number d, is called an arithmetic progression, and the number d is the difference of an arithmetic progression.

Thus, an arithmetic progression is a numerical sequence (an) given recursively by the relations

a1 = a, an = an – 1 + d (n = 2, 3, 4, ...)

A geometric progression is a sequence, all members of which are nonzero and each term of which, starting from the second, is obtained from the previous term by multiplying by the same number q.

Thus, a geometric progression is a numerical sequence (bn) given recursively by the relations

b1 = b, bn = bn – 1 q (n = 2, 3, 4…).

1.1 Types of sequences.

1.1.1 Limited and Unlimited Sequences.

A sequence (bn) is called bounded from above if there is a number M such that for any number n the inequality bn≤ M is satisfied;

A sequence (bn) is called bounded from below if there is a number M such that for any number n the inequality bn≥ M is satisfied;

For example:

1.1.2 Monotonicity of sequences.

A sequence (bn) is called non-increasing (non-decreasing) if, for any number n, the inequality bn≥ bn + 1 (bn ≤bn + 1) is true;

A sequence (bn) is called decreasing (increasing) if, for any number n, the inequality bn> bn + 1 (bn

Decreasing and increasing sequences are called strictly monotone, non-increasing monotone in the broad sense.

Sequences that are bounded at the top and bottom at the same time are called bounded.

The sequence of all these types are collectively called monotonic.

1.1.3 Infinitely large and small sequences.

An infinitesimal sequence is a numeric function or sequence that tends to zero.

A sequence an is called infinitesimal if

A function is called infinitesimal in a neighborhood of the point x0 if ℓimx → x0 f (x) = 0.

A function is called infinitesimal at infinity if ℓimx →. + ∞ f (x) = 0 or ℓimx → -∞ f (x) = 0

Also, an infinitesimal function is the difference between a function and its limit, that is, if ℓimx →. + ∞ f (x) = a, then f (x) - a = α (x), ℓimx →. + ∞ f (( x) -a) = 0.

An infinitely large sequence is a numerical function or a sequence that tends to infinity.

A sequence an is called infinitely large if

ℓimn → 0 an = ∞.

A function is called infinitely large in a neighborhood of the point x0 if ℓimx → x0 f (x) = ∞.

A function is called infinitely large at infinity if

ℓimx →. + ∞ f (x) = ∞ or ℓimx → -∞ f (x) = ∞.

1.1.4 Properties of infinitesimal sequences.

The sum of two infinitesimal sequences is itself also an infinitesimal sequence.

The difference of two infinitesimal sequences is itself also an infinitesimal sequence.

Algebraic sum of any finite number infinitesimal sequences are themselves also infinitesimal sequences.

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

The product of any finite number of infinitesimal sequences is an infinitesimal sequence.

Any infinitesimal sequence is limited.

If a stationary sequence is infinitesimal, then all its elements, starting with some one, are equal to zero.

If the whole infinitesimal sequence consists of identical elements, then these elements are zeros.

If (xn) is an infinitely large sequence that does not contain zero terms, then there is a sequence (1 / xn) that is infinitely small. If, nevertheless, (xn) contains zero elements, then the sequence (1 / xn) can still be defined, starting from some number n, and will still be infinitely small.

If (an) is an infinitely small sequence that does not contain zero terms, then there is a sequence (1 / an) that is infinitely large. If, nevertheless, (an) contains zero elements, then the sequence (1 / an) can still be defined starting from some number n, and will still be infinitely large.

1.1.5 Converging and diverging sequences and their properties.

A converging sequence is a sequence of elements of a set X that has a limit in this set.

A divergent sequence is a sequence that is not convergent.

Any infinitesimal sequence is convergent. Its limit is zero.

Removing any finite number of elements from an infinite sequence does not affect either the convergence or the limit of this sequence.

Any converging sequence is bounded. However, not every limited sequence converges.

If the sequence (xn) converges, but is not infinitesimal, then, starting from some number, the sequence (1 / xn) is defined, which is bounded.

The sum of the converging sequences is also a converging sequence.

The difference of the converging sequences is also a converging sequence.

The product of converging sequences is also a converging sequence.

The quotient of two converging sequences is defined starting from some element, unless the second sequence is infinitesimal. If the quotient of two converging sequences is defined, then it is a converging sequence.

If a converging sequence is bounded from below, then none of its lower bounds exceeds its limit.

If a converging sequence is bounded from above, then its limit does not exceed any of its upper bounds.

If for any number the members of one converging sequence do not exceed the members of another converging sequence, then the limit of the first sequence also does not exceed the limit of the second.

Lecture 8. Numerical sequences.

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

– abbreviated notation
, (8.1)

will callnumerical sequence or just a sequence.

Separate 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 recursive way of specifying a sequence is used: the first few members of the sequence are given and a formula is used to calculate the next elements. For example, the 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
for any
... We get a sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34,…. For such a series, it is rather difficult to find a formula for the general term.

8.1. Arithmetic operations with sequences.

Consider two sequences:

(8.1)

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

Let's call the sequence sum of sequences (8.1) and (8.2), we write as follows:; 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. Limited and unlimited sequences.

The collection of all elements in an arbitrary sequence
forms some numerical set, which 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 from above , if ; M  top edge.

Definition 8.4. Subsequence
calledlimited from 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 the condition of the 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 (arbitrarily 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, since limited only from below.

8.3. Infinitely large and infinitely small sequences.

Definition 8.7. Subsequence
calledinfinitely large if for any (arbitrarily large) real number A there is a number
such that for all
the elementsx n
.

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

 Example 8.1.
is infinitely large. Take any number A> 0. From the inequality
we get n>A... If you take
then for all n>N the inequality
, that is, according to Definition 8.7, the sequence
infinitely large. 

Definition 8.8. Subsequence
calledinfinitesimal if for
(however small ) there is a number
such that for all
the elements of this sequence satisfy the inequality
.

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

Take any number
... From the inequality
we get ... If you take
then for all n>N the inequality
. 

♦ Statement 8.2. Subsequence
is infinitely large for
and infinitely small for
.

Proof.

1) Let first
:
, where
... By the Bernoulli formula (Example 6.3, p. 6.1.)
... We fix an arbitrary positive number A and select a number by it N such that the inequality is true:

,
,
,
.

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

.

Thus, for
there is such a number
that for all

- infinitely large at
.

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

Let be
, where
, by the Bernoulli formula
or
.

We fix
,
and choose
such that

,
,
.

For

... We indicate such a number N that for all

, that is, for
subsequence
infinitely small. ■

8.4. Basic properties of infinitesimal sequences.

♦ Theorem 8.1.Sum

and

Proof. We fix ;
- infinitely small

,

- infinitely small

... Let's choose
... Then at

,
,
. ■

♦ Theorem 8.2. Difference
two infinitesimal sequences
and
there is an infinitely small sequence.

For proof of the theorem, it suffices 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 by an infinitesimal sequence is an infinitesimal sequence.

Proof.
- limited,
- an infinitely small sequence. We fix ;
,
;
: at
fair
... Then
. ■

♦ Theorem 8.4.Any infinitesimal sequence is bounded.

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

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

♦ Theorem 8.5.

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

Proof 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 infinitely small sequence.