Number sequence. How to calculate the limits of sequences? What is a neighborhood

Vida y= f(x), x ABOUT N, Where N– many natural numbers(or natural argument function), denoted y=f(n) or y 1 ,y 2 ,…, y n,…. Values y 1 ,y 2 ,y 3 ,… are called respectively the first, second, third, ... members of the sequence.

For example, for the function y= n 2 can be written:

y 1 = 1 2 = 1;

y 2 = 2 2 = 4;

y 3 = 3 2 = 9;…y n = n 2 ;…

Methods for specifying sequences. Sequences can be specified in various ways, among which three are especially important: analytical, descriptive and recurrent.

1. A sequence is given analytically if its formula is given n th member:

y n=f(n).

Example. y n= 2n – 1 – sequence of odd numbers: 1, 3, 5, 7, 9, …

2. Descriptive The way to specify a numerical sequence is to explain from which elements the sequence is built.

Example 1. “All terms of the sequence are equal to 1.” This means, we're talking about about the stationary sequence 1, 1, 1, …, 1, ….

Example 2: “The sequence consists of all prime numbers in ascending order.” Thus, the given sequence is 2, 3, 5, 7, 11, …. With this method of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to.

3. The recurrent method of specifying a sequence is to specify a rule that allows you to calculate n-th member of a sequence if its previous members are known. The name recurrent method comes from the Latin word recurrent- come back. Most often, in such cases, a formula is indicated that allows one to express n th member of the sequence through the previous ones, and specify 1–2 initial members of the sequence.

Example 1. y 1 = 3; y n = y n–1 + 4 if n = 2, 3, 4,….

Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; ….

You can see that the sequence obtained in this example can also be specified analytically: y n= 4n – 1.

Example 2. y 1 = 1; y 2 = 1; y n = y n –2 + y n–1 if n = 3, 4,….

Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8;

The sequence in this example is especially studied in mathematics because it has a number of interesting properties and applications. It is called the Fibonacci sequence, named after the 13th century Italian mathematician. It is very easy to define the Fibonacci sequence recurrently, but very difficult analytically. n The th Fibonacci number is expressed through its serial number by the following formula.

At first glance, the formula for n the th Fibonacci number seems implausible, since the formula that specifies the sequence of natural numbers alone contains square roots, but you can check “manually” the validity of this formula for the first few n.

Properties of number sequences.

Number sequence – special case numerical function, therefore a number of properties of functions are also considered for sequences.

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

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

Definition.Sequence ( y n} is called decreasing if each of its terms (except the first) is less than the previous one:

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

Increasing and decreasing sequences are combined under the common term - monotonic sequences.

Example 1. y 1 = 1; y n= n 2 – increasing sequence.

Thus, the following theorem is true (a characteristic property of an arithmetic progression). A number sequence is arithmetic if and only if each of its members, except 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 numbers 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.

Solving this equation gives x= –5,5. At this value x given expressions 3 x + 2, 5x– 4 and 11 x+ 12 take, respectively, the values –14.5, –31,5, –48,5. This - arithmetic progression, its difference is –17.

Geometric progression.

A numerical sequence, all of whose terms are non-zero and each of whose terms, starting from the second, is obtained from the previous term by multiplying by the same number q, called geometric progression, and the number q- the denominator of a geometric progression.

Thus, a geometric progression is a number 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 increasing sequence if b 1 > 0, q> 1, and decreasing if b 1 > 0, 0 q

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

b 1 2 , b 2 2 , b 3 2 , …, b n 2,... is a geometric progression whose first term is equal to b 1 2 , and the denominator is q 2 .

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

b n= b 1 qn– 1 .

You can obtain a formula for the sum of terms of a finite geometric progression.

Let a finite geometric progression be given

b 1 ,b 2 ,b 3 , …, b n

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

S n= b 1 + b 2 + b 3 + … +b n.

It is accepted that q No. 1. To determine S n an artificial technique is used: 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 the formula with umma n terms of geometric progression for the case when q≠ 1.

At q= 1 the formula need not be derived separately; it is obvious that in this case S n= a 1 n.

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

bn=bn- 1 q;

bn = bn+ 1 /q,

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

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

Consistency limit.

Let there be a sequence ( c n} = {1/n}. This sequence is called harmonic, since each of its terms, starting from the second, is the harmonic mean between the previous and subsequent terms. Average geometric 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 a limit A=0 for the harmonic sequence ( c n} = {1/n). Let ε be arbitrarily small positive number. The difference is considered

Does such a thing exist? N that's for everyone n ≥ N inequality 1 holds /N ? If we take it as N any natural number greater than 1/ε , then for everyone n ≥ N inequality 1 holds /n ≤ 1/N ε , Q.E.D.

Proving the presence of a limit for a particular sequence can sometimes be very difficult. The most frequently occurring sequences are well studied and are listed in reference books. There are important theorems that allow you to conclude that a given sequence has a limit (and even calculate it), based on already studied sequences.

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

Theorem 2. If a sequence is monotonic 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+ c) and (| a n|} have limits cA, A +c, |A| accordingly (here c– arbitrary number).

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

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

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

Anna Chugainova

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.

Types of sequence

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

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

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

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

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

Sequence example

Let's check whether 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. This means the sequence (n-1)/n is increasing.

Let X (\displaystyle X) is either a set of real numbers R (\displaystyle \mathbb (R) ), or a set 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

Operations on sequences

Subsequences

Subsequence sequences (x n) (\displaystyle (x_(n)))- this is a 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

A sequence of prime numbers is a subsequence of a sequence of natural numbers.
The sequence of natural numbers, multiples of , is a subsequence of the sequence of even natural numbers.

Properties

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

Sequence limit

Sequence limit - this is an object to which the members of the sequence approach as the number increases. Thus, in an arbitrary topological space, the limit of a sequence is an element in any neighborhood of which all members of the sequence, starting from a certain point, lie. In particular, for number sequences, a limit is a number in any neighborhood of which all terms of the sequence starting from a certain point lie.

Fundamental Sequences

Fundamental Sequence (convergent sequence , Cauchy sequence ) is a sequence of elements of a metric space in which, for any given distance in advance, there is an element whose distance to any of the following elements does not exceed the given one. For number sequences, the concepts of fundamental and convergent sequences are equivalent, but in general this is not the case.