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 Limit 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 informative topic. This topic is found in tasks of increased complexity, which are offered to students by authors of didactic materials, in problems of mathematical olympiads, entrance exams to Higher Educational Institutions and the Unified State Exam. I am interested in learning the relationship of mathematical sequences with other areas of knowledge.

The purpose of the research work: To expand knowledge about 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 increasing 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 numeric function or 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.

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

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 entire 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.

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

The numerical sequence can be specified 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 sequentially 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) is called 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, called an unbounded sequence.

increasing, if the following equality X (n + 1)> Xn holds for all natural n. In other words, each member of the sequence, starting with the second, must be larger than the previous member.

The sequence (Xn) is called decreasing if for all natural 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.

If each natural number n is associated with some real number x n, then they say that given numerical sequence

x 1 , x 2 , … x n , …

Number x 1 is called a member of the sequence with number 1 or the first member of the sequence, number x 2 - a member of the sequence with number 2 or the second member of the sequence, etc. The number x n is called member of the sequence numbered n.

There are two ways to set number sequences - with and with recurrent formula.

Sequencing with common term formulas Is a sequence assignment

x 1 , x 2 , … x n , …

using a formula expressing the dependence of the term x n on its number n.

Example 1. Numerical sequence

1, 4, 9, … n 2 , …

given using the common term formula

x n = n 2 , n = 1, 2, 3, …

Sequencing using a formula expressing a member of the sequence x n in terms of the sequence members with preceding numbers is called sequencing using recurrent formula.

x 1 , x 2 , … x n , …

are called increasing sequence, more the preceding member.

In other words, for everyone n

x n + 1 >x n

Example 3. Sequence of natural numbers

1, 2, 3, … n, …

is an increasing sequence.

Definition 2. Number sequence

x 1 , x 2 , … x n , …

are called decreasing sequence, if each member of this sequence smaller the preceding member.

In other words, for everyone n= 1, 2, 3, ... the inequality

x n + 1 < x n

Example 4. Subsequence

given by the formula

is an descending sequence.

Example 5. Numerical sequence

1, - 1, 1, - 1, …

given by the formula

x n = (- 1) n , n = 1, 2, 3, …

is not neither increasing nor decreasing sequence.

Definition 3. Increasing and decreasing numerical sequences are called monotonic sequences.

Limited and unlimited sequences

Definition 4. Number sequence

x 1 , x 2 , … x n , …

are called bounded from above, if there is a number M such that each member of this sequence smaller numbers M.

In other words, for everyone n= 1, 2, 3, ... the inequality

Definition 5. Numerical sequence

x 1 , x 2 , … x n , …

are called bounded from below, if there is a number m such that each member of this sequence more numbers m.

In other words, for everyone n= 1, 2, 3, ... the inequality

Definition 6. Number sequence

x 1 , x 2 , … x n , …

called limited if it bounded both above and below.

In other words, there are numbers M and m such that for all n= 1, 2, 3, ... the inequality

m< x n < M

Definition 7. Numerical sequences that are not limited are called unlimited sequences.

Example 6. Numerical sequence

1, 4, 9, … n 2 , …

given by the formula

x n = n 2 , n = 1, 2, 3, … ,

bounded from below, for example, the number 0. However, this sequence unlimited from above.

Example 7. Subsequence

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 for calculating the following elements is given. 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 the sum of the sequences (8.1) and (8.2), we write it 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
calledbounded 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 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 infinitesimal for
.

Proof.

1) Let first
:
, where
... By the Bernoulli formula (Example 6.3, Section 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 infinitesimal sequence.

2) If all elements of an infinitesimal sequence
are nonzero, 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 - infinitely small.

2) Let
- an infinitely small sequence. Suppose all elements
are nonzero. We fix A;
or
at
... By definition 8.7, the sequence infinitely large. ■

Let be X (\ displaystyle X) is either a 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)) elements of the set X (\ displaystyle X) called numerical sequence.

Examples of

Sequence operations

Subsequences

Subsequence sequences (x n) (\ displaystyle (x_ (n))) is the sequence (x n k) (\ displaystyle (x_ (n_ (k)))), where (n k) (\ displaystyle (n_ (k)))- 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 of

• A sequence of primes is a subsequence of a sequence of natural numbers.
• A sequence of multiples of natural numbers is a subsequence of a sequence of even natural numbers.

Properties

Limit point of sequence is a point, in any neighborhood of which there are infinitely many elements of this sequence. For converging number sequences, the limit point is the same as the limit.

Sequence limit

Sequence limit is an object that the members of the sequence approach with increasing number. So, in an arbitrary topological space, the limit of a sequence is an element in any neighborhood of which all the members of the sequence lie, starting with 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 (converging sequence , Cauchy sequence ) is a sequence of elements of metric space, in which for any predetermined distance there is such an element, the distance from which to any of the following elements does not exceed a given one. For numerical sequences, the concepts of fundamental and convergent sequences are equivalent, but in general this is not the case.