How to calculate sequence limits? Numeric Sequences A specific sequence of numbers.

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

If the function is given on the set of natural numbers
, then the set of values of the function will be countable and each number
number is matched
. In this case, we say that given numerical sequence. Numbers are called elements or members of a sequence, and the number - general or -th member of the sequence. Each element has a follower
. This explains the use of the term "sequence".

The sequence is usually specified either by listing its elements, or by indicating the law by which the element with the number is calculated , i.e. indicating the formula th member .

Example.Subsequence
can be given by the formula:
.

Usually sequences are denoted as follows: etc., where the formula of its th member.

Example.Subsequence
‑this is the sequence

The set of all elements of a sequence
denoted
.

Let
And
- two sequences.

WITH ummah sequences
And
call the sequence
, Where
, i.e..

R aznosti of these sequences is called the sequence
, Where
, i.e..

If And ‑ constants, then the sequence
,
called linear combination sequences
And
, i.e.

work sequences
And
call the sequence -th member
, i.e.
.

If
, then it is possible to determine 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 product and the quotient are equal
.

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

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

Converging Sequences

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

Number called sequence limit
. At the same time, they record
or
.

Example.
.

Let us show that
. Set any number
. Inequality
performed for
, such that
that the definition of convergence holds 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
(when) the elements of the sequence are in the interval
, which is called -neighborhood of the point .

Subsequence
, whose limit is equal to zero (
, or
at
) is called infinitesimal.

As applied to infinitesimals, the following statements are true:

The sum of two infinitesimals is infinitesimal;

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

Theorem .In order for the sequence
had a limit, it is necessary and sufficient that
, Where - constant; - infinitely small .

Main properties of convergent sequences:

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

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

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

– abbreviated notation
, (8.1)

we will callnumerical sequence or just a sequence.

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

The sequence can be specified by a general term formula, like so:
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 way of specifying a sequence is used: the first few members of the sequence and a formula for calculating the next elements are given. For example, the sequence defined by the first element and the recurrence relation
(arithmetic progression). Consider a sequence called near Fibonacci: set the first two elements x 1 =1, x 2 =1 and the 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 a common term.

8.1. Arithmetic operations with sequences.

Consider two sequences:

(8.1)

Definition 8.2. Let's callthe product 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 as follows: ; likewise
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 numerical 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 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- top and bottom edges
.

Inequalities (8.3) are called sequence boundedness condition
.

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, because 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
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
is not satisfied for all even n. ☼

 Example 8.1.
is infinitely large. Take any number A>0. From inequality
we get n>A. If you take
, then for all n>N inequality will hold
, 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
elements this sequence satisfy the inequality
.

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

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

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

Proof.

1) Let first
:
, Where
. According to the Bernoulli formula (Example 6.3, Section 6.1.)
. We fix an arbitrary positive number A and choose a number N such that the inequality is true:

,
,
,
.

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

.

So for
there is a number
, that for all

- infinitely large
.

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

Let
, Where
, according to the Bernoulli formula
or
.

Fixing
,
and choose
such that

,
,
.

For

. Specify this number N, that for all

, that is, when
subsequence
infinitely small. ■

8.4. Basic properties of infinitesimal sequences.

♦ Theorem 8.1.Sum

And

Proof. Fixing ;
- infinitely small

,

- infinitely small

. Let's choose
. Then at

,
,
. ■

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

For proof 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 and an infinitesimal sequence is an infinitesimal sequence.

Proof.
- limited
is an infinitesimal sequence. Fixing ;
,
;
: at
fair
. Then
. ■

♦ Theorem 8.4.Every infinitesimal sequence is bounded.

Proof. Fixing Let some number . Then
for all rooms n, which means that the sequence is bounded. ■

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
are 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 different from zero, then the quotient two sequences
And
is an infinite sequence.

Proof.

1) Let
is an infinitely large sequence. Fixing ;
or
at
. Thus, by definition 8.8, the sequence - infinitely small.

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

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

Examples

Operations on sequences

Subsequences

Subsequence sequences (x n) (\displaystyle (x_(n))) is the sequence (x n k) (\displaystyle (x_(n_(k)))), Where (n k) (\displaystyle (n_(k))) is 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

The sequence of prime numbers is a subsequence of the sequence of natural numbers.
The sequence of natural numbers that are 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 numerical sequences, the limit point coincides with the limit.

Sequence limit

Sequence limit is the object that 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 lie, starting from 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 (self-convergent sequence , Cauchy sequence ) is a sequence of elements of a metric space , in which, for any predetermined distance, there is such an element, the distance from which to any of the elements following it does not exceed the given one. For numerical sequences, the concepts of fundamental and convergent sequences are equivalent, but in the general case this is not the case.

Vida y= f(x), x ABOUT N, Where N is the set of natural numbers (or a function of a natural argument), 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 setting 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 that it explains what elements the sequence is built from.

Example 1. "All members of the sequence are equal to 1." This means that we are talking about a stationary sequence 1, 1, 1, …, 1, ….

Example 2. "The sequence consists of all prime numbers in ascending order." Thus, the sequence 2, 3, 5, 7, 11, … is given. 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 way of specifying a sequence is that a rule is indicated that allows one to calculate n-th member of the sequence, if its previous members are known. The name recurrent method comes from the Latin word recurrere- come back. Most often, in such cases, a formula is indicated that allows expressing 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; ….

It can be seen 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 composed in this example is specially studied in mathematics because it has a number of interesting properties and applications. It is called the Fibonacci sequence - after the Italian mathematician of the 13th century. Defining the Fibonacci sequence recursively is very easy, but analytically it is very difficult. n The th Fibonacci number is expressed in terms of its ordinal number by the following formula.

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

Properties of numerical sequences.

A numerical sequence is a special case of a numerical function, so 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 united by a common term - monotonic sequences.

Example 1 y 1 = 1; y n= n 2 is an increasing 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 last in the case of a finite sequence), is equal to the arithmetic mean of the previous 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.

Solving this equation gives x= –5,5. With 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 is an arithmetic progression, its difference is -17.

Geometric progression.

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

Thus, a geometric progression is a numerical sequence ( b n) given 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, 0q

One of the obvious properties of a geometric progression is that if a 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- th term of a geometric progression has the form

b n= b 1 q n– 1 .

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

Let there be a finite geometric progression

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 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 hence

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

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

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

b n = b n- 1 q;

bn = bn+ 1 /q,

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

a numerical 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.

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 a limit A=0 for the harmonic sequence ( c n} = {1/n). Let ε be an arbitrarily small positive number. We consider the difference

Is there such N that for everyone n≥ N inequality 1 /N? If taken as N any natural number greater than 1/ε , then for all n ≥ N inequality 1 /n ≤ 1/N ε , Q.E.D.

It is sometimes very difficult to prove the existence of a limit for a particular sequence. The most common sequences are well studied and are listed in reference books. There are important theorems that make it possible 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 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+ c) and (| a n|} have limits cA, A +c, |A| respectively (here c is 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 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 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 element of the sequence following it.

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

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

In mathematics, many different sequences are considered:

time series of both numerical and non-numerical nature;
sequences of elements of a metric space
sequences of function 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 of elements of arbitrary nature be given. | Any mapping of the set of natural numbers into 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 the sequence member 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 numerical sequence.

Notation

Sequences of the form

It is customary to write compactly using parentheses:

curly braces are sometimes used:

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

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

How to calculate sequence limits? Numeric Sequences A specific sequence of numbers.

Converging Sequences

Examples

Operations on sequences

Subsequences

Examples

Properties

Sequence limit

Fundamental sequences

Properties of numerical sequences.

Geometric progression.

Sequence limit.

Definition

Related definitions

Comments

Notation

see also

See what "Sequence" is in other dictionaries:

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