If everyone natural number n is assigned to some real number x n , then they say that it is given number sequence
x 1 , x 2 , … x n , …
Number x 1 is called a member of the sequence with number 1 or first term of the sequence, number x 2 - 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 with number n.
There are two ways to specify number sequences - with and with recurrent formula.
Sequence using formulas for the general term of a sequence– this is a sequence task
x 1 , x 2 , … x n , …
using a formula expressing the dependence of the term x n on its number n.
Example 1. Number sequence
1, 4, 9, … n 2 , …
given using the common term formula
x n = n 2 , n = 1, 2, 3, …
Specifying a sequence using a formula expressing a sequence member x n through the sequence members with preceding numbers is called specifying a sequence using recurrent formula.
x 1 , x 2 , … x n , …
called in increasing sequence, more previous member.
In other words, for everyone n
x n + 1 >x n
Example 3. Sequence of natural numbers
1, 2, 3, … n, …
is ascending sequence.
Definition 2. Number sequence
x 1 , x 2 , … x n , …
called descending sequence if each member of this sequence less previous member.
In other words, for everyone n= 1, 2, 3, … the inequality is satisfied
x n + 1 < x n
Example 4. Subsequence
given by the formula
is descending sequence.
Example 5. Number 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 number sequences are called monotonic sequences.
Bounded and Unbounded Sequences
Definition 4. Number sequence
x 1 , x 2 , … x n , …
called bounded above, if there is a number M such that each member of this sequence less numbers M.
In other words, for everyone n= 1, 2, 3, … the inequality is satisfied
Definition 5. Number sequence
x 1 , x 2 , … x n , …
called bounded 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 is satisfied
Definition 6. Number sequence
x 1 , x 2 , … x n , …
is called limited if it limited both above and below.
In other words, there are numbers M and m such that for all n= 1, 2, 3, … the inequality is satisfied
m< x n < M
Definition 7. Numeric sequences that are not limited, called unlimited sequences.
Example 6. Number sequence
1, 4, 9, … n 2 , …
given by the formula
x n = n 2 , n = 1, 2, 3, … ,
bounded below, for example, the number 0. However, this sequence unlimited from above.
Example 7. Subsequence
given by the formula
is limited sequence , because for everyone n= 1, 2, 3, … the inequality is satisfied
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Vida y= f(x), x ABOUT N, Where N– a 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 specifying sequences. Sequences can be specified different 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 way of setting number sequence is that it explains 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. “A 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 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 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 composed in this example is specially studied in mathematics, since 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 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 descending sequences are combined general 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 terms except the first (and the last in the case finite sequence), is equal to the arithmetic mean of the preceding and subsequent terms.
Example. At what value x numbers 3 x + 2, 5x– 4 and 11 x+ 12 form a finite arithmetic progression?
According to 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 the obvious properties of a geometric progression is that if the sequence is a geometric progression, then so is 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 applies artificial reception: some are executed geometric transformations expressions 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), equal to the product previous and subsequent members.
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
IN 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. Available important theorems, allowing one to draw a conclusion about the presence of a limit for a given sequence (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
The definition of a numerical sequence is given. Examples of infinitely increasing, convergent and divergent sequences are considered. A sequence containing all rational numbers is considered.
Definition .
Numerical sequence (xn)
is a law (rule) according to which, for every natural number n = 1, 2, 3, . . .
a certain number x n is assigned.
The element x n is called nth term or an element of a sequence.
The sequence is denoted as the nth term enclosed in curly braces: . The following designations are also possible: . They explicitly indicate that the index n belongs to the set of natural numbers and the sequence itself has an infinite number of terms. Here are some example sequences:
,
,
.
In other words, a number sequence is a function whose domain of definition is the set of natural numbers. The number of elements of the sequence is infinite. Among the elements there may also be members having same values. Also, a sequence can be considered as a numbered set of numbers consisting of an infinite number of members.
We will be mainly interested in the question of how sequences behave when n tends to infinity: . This material is presented in the section Limit of a sequence - basic theorems and properties. Here we will look at some examples of sequences.
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 indefinitely in absolute value, but don't have 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
. Members with odd numbers:
,
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 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 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 converging to different meanings, then 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.
A rational number r can be represented in the following form:
,
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 with will correspond 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 top part the following 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 converging to different numbers, then 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. Precise definition convergence is discussed on the page Determining the Limit of a Sequence. Related properties and theorems are stated on the page
The concept of a number sequence.
Let each natural number n correspond to a number a n , then we say that a function a n =f(n) is given, which is called a number sequence. Denoted by a n ,n=1,2,… or (a n ).
The numbers a 1 , a 2 , ... are called members of the sequence or its elements, a n is the general member of the sequence, n is the number of the member a n .
By definition, any sequence contains infinite set elements.
Examples of number sequences.
Arithmetic progression – numerical progression of the form:
that is, a sequence of numbers (terms of the progression), each of which, starting from the second, is obtained from the previous one by adding to it a constant number d (step or difference of the progression): 
.
Any term of the progression can be calculated using the general term formula:
Any member of an arithmetic progression, starting from the second, is the arithmetic mean of the previous and next members of the progression: 
The sum of the first n terms of an arithmetic progression can be expressed by the formulas: 
The sum of n consecutive terms of an arithmetic progression starting with term k: 
An example of the sum of an arithmetic progression is the sum of a series of natural numbers up to n inclusive:
Geometric progression - sequence of numbers 
(members of a progression), in which each subsequent number, starting from the second, is obtained from the previous one by multiplying it by a certain number q (denominator of the progression), where 
,
:
Any term of a geometric progression can be calculated using the formula: 
If b 1 > 0 and q > 1, the progression is an increasing sequence if 0 The progression got its name from its characteristic property: The product of the first n terms of a geometric progression can be calculated using the formula: The product of the terms of a geometric progression starting with the k-th term and ending with the n-th term can be calculated using the formula: Sum of the first n terms of a geometric progression: If Consistency limit. A sequence is called increasing if each member is greater than the previous one. A sequence is called decreasing if each member is less than the previous one. A sequence x n is called bounded if there are numbers m and M such that for any natural number n the condition is satisfied It may happen that all members of the sequence (a n ) with unlimited growth of the number n will approach some number m. A number a is called the limit of the sequence X n if for every Ε>0 there is a number (depending on Ε) n 0 =n o (Ε) such that for In this case they write Convergence of sequences. A sequence whose limit is finite is said to converge to a: If a sequence does not have a finite (countable) limit, it will be called divergent. Geometric meaning. If Theorem 1) On the uniqueness of the limit: If the sequence converges, that is, has a limit, then this limit is unique. Theorem 2) If the sequence a n converges to a: Theorem 3) Prerequisite existence of a limit. If a sequence converges, that is, has a limit, then it is bounded. Proof: let’s select n>N such that: Theorem 4) Sufficient condition for the existence of a limit. If a sequence is monotonic and bounded, then it has a limit. . Theorem 5) Let Three sequence theorem. If Limit Properties. If (xn) and (yn) have limits, then: Limit of ratio of polynomials (fractions). Let x n and y n be polynomials in degree k, respectively, that is: x n =P k (n)=a 0 n k +a 1 n k-1 +…+a k , y n =Q m (n)=b 0 n m +b 1 n m-1 +…+b m The limit of the ratio of polynomials is equal to the limit of the ratio of their leading terms: If the degree of the numerator is equal to the degree of the denominator, then the limit is equal to the ratio of the coefficients at higher degrees. If the degree of the numerator is less than the degree of the denominator, the limit is zero. If the degree of the numerator is greater than the degree of the denominator, the limit tends to infinity.
that is, each term is equal to the geometric mean of its neighbors.
, then when 
, And
at 
.
.
inequality holds 
for all (natural)n>n 0 .
or 
.
, then all members of this sequence, with the exception of the last number, will fall into an arbitrary Ε neighborhood of point a. Geometrically, the boundedness of a sequence means that all its values lie on a certain segment.
, then any subsequence of it 
has the same limit.
∎
and let the condition x n ≤y n be satisfied for any n, thena 
and for sequences x n ,y n ,z n the condition x n ≤y n ≤z n is satisfied, then for 
should 
.
