Numerical sequence. How to find the limit of a sequence.

Consider the series natural numbers: 1, 2, 3, , n – 1, n,  .

If we replace every natural number n in this series some number a n, following some law, we get a new series of numbers:

a 1 , a 2 , a 3 , , a n –1 , a n , ,

abbreviated and called numerical sequence. Value a n is called the common member of the numerical sequence. Usually the numerical sequence is given by some formula a n = f(n) that allows you to find any member of the sequence by its number n; this formula is called the general term formula. Note that it is not always possible to specify a numerical sequence by a general term formula; sometimes a sequence is specified by describing its members.

By definition, a sequence always contains an infinite number of elements: any two different elements of it differ at least in their numbers, of which there are infinitely many.

The numeric sequence is a special case of a function. A sequence is a function defined on the set of natural numbers and taking values in the set of real numbers, i.e., a function of the form f : N  R.

Subsequence
called increasing(waning), if for any n  N
Such sequences are called strictly monotonous.

Sometimes it is convenient to use as numbers not all natural numbers, but only some of them (for example, natural numbers starting from some natural number n 0). For numbering, it is also possible to use not only natural numbers, but also other numbers, for example, n= 0, 1, 2,  (here, zero is added to the set of natural numbers as another number). In such cases, specifying the sequence, indicate what values the numbers take. n.

If in some sequence for any n  N
then the sequence is called non-decreasing(non-increasing). Such sequences are called monotonous.

Example 1 . Numeric sequence 1, 2, 3, 4, 5, ... is a series of natural numbers and has a common term a n = n.

Example 2 . The number sequence 2, 4, 6, 8, 10, ... is a series of even numbers and has a common term a n = 2n.

Example 3 . 1.4, 1.41, 1.414, 1.4142, … - numerical sequence of approximate values with increasing accuracy.

In the last example, it is impossible to give a formula for the common term of the sequence.

Example 4 . Write the first 5 terms of a numerical sequence by its common term
. To calculate a 1 is needed in the formula for the common term a n instead of n substitute 1 to calculate a 2 − 2, etc. Then we have:

Test 6 . The common member of the sequence 1, 2, 6, 24, 120,  is:

Test 7 .
is:

Test 8 . Common Member of the Sequence
is:

Number Sequence Limit

Consider a numerical sequence whose common term approaches a certain number BUT with increasing serial number n. In this case, the number sequence is said to have a limit. This concept has a more rigorous definition.

Number BUT is called the limit of the number sequence
:

(1)

if for any  > 0 there is such a number n 0 = n 0 (), depending on , which
at n > n 0 .

This definition means that BUT there is a limit of a number sequence if its common term indefinitely approaches BUT with increasing n. Geometrically, this means that for any  > 0 one can find such a number n 0 , which, starting from n > n 0 , all members of the sequence are located inside the interval ( BUT – , BUT+ ). A sequence that has a limit is called converging; otherwise - divergent.

A number sequence can have only one limit (finite or infinite) of a certain sign.

Example 5 . Harmonic sequence has the number 0 as a limit. Indeed, for any interval (–; +) as a number N 0 can be any integer greater than . Then for all n > n 0 > we have

Example 6 . The sequence 2, 5, 2, 5,  is divergent. Indeed, no interval of length less, for example, one, can contain all members of the sequence, starting from some number.

The sequence is called limited if there is such a number M, what
for all n. Every convergent sequence is bounded. Every monotone and bounded sequence has a limit. Every convergent sequence has a unique limit.

Example 7 . Subsequence
is increasing and limited. She has a limit
=e.

Number e called Euler number and is approximately equal to 2.718 28.

Test 9 . The sequence 1, 4, 9, 16,  is:

1) converging;

2) divergent;

3) limited;

Test 10 . Subsequence
is:

1) converging;

2) divergent;

3) limited;

4) arithmetic progression;

5) geometric progression.

Test 11 . Subsequence is not:

1) converging;

2) divergent;

3) limited;

4) harmonic.

Test 12 . Limit of the sequence given by the common term
equal.

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 ( x n ) called the law (rule), according to which, for each natural number n = 1, 2, 3, . . . some number x n is assigned.
The element x n is called nth member or an element of a sequence.

The sequence is denoted as the nth member enclosed in curly brackets: . The following designations are also possible: . They explicitly state that the index n belongs to the set of natural numbers and that the sequence itself has an infinite number of members. Here are some examples of sequences:
, , .

In other words, a numerical sequence is a function whose domain is the set of natural numbers. The number of elements in the sequence is infinite. Among the elements, there may also be members that have the same value. Also, the 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 - how sequences behave when n tends to infinity: . This material is presented in the Sequence limit - basic theorems and properties section. And here we will look at some examples of sequences.

Sequence examples

Examples of infinitely increasing sequences

Let's consider a sequence. The general term of this sequence is . Let's write out the first few terms:
.
It can be seen that as the number n grows, the elements increase indefinitely towards positive values. We can say that this sequence tends to : at .

Now consider a sequence with a common term . Here are some of its first members:
.
As the number n grows, the elements of this sequence increase in absolute value indefinitely, but do not have a constant sign. That is, this sequence tends to : at .

Examples of sequences converging to a finite number

Let's consider a sequence. Its common member The first terms are as follows:
.
It can be seen that as the number n grows, the elements of this sequence approach their limit value a = 0 : at . So each subsequent term is closer to zero than the previous one. In a sense, we can assume that there is an approximate value for the number a = 0 with an error. It is clear that as n grows, this error tends to zero, that is, by choosing n, the error can be made arbitrarily small. Moreover, for any given error ε > 0 it is possible to specify such a number N , 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. Its common member Here are some of its first members:
.
In this sequence, even-numbered terms are zero. Members with odd n are . Therefore, as n grows, their values approach the limiting value a = 0 . This also follows from the fact that
.
As in the previous example, we can specify an arbitrarily small error ε > 0 , for which it is possible to find such a number N that elements with numbers greater than N will deviate from the limit value a = 0 by a value 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 the 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 consider a more interesting sequence. Take a segment on the number line. Let's split it in half. We get two segments. Let
.
Each of the segments is again divided in half. We get four segments. Let
.
Divide each segment in half again. Let's take

.
And so on.

As a result, we obtain a sequence whose elements are distributed in an open interval (0; 1) . Whatever point we take from the closed interval , we can always find members of the sequence that are arbitrarily close to this point, or coincide with it.

Then from the original sequence one can single out a subsequence that will converge to an arbitrary point from the interval . That is, as the number n grows, the members of the subsequence will come closer and closer to the preselected point.

For example, for point a = 0 you can choose the following subsequence:
.
= 0 .

For point a = 1 choose the following subsequence:
.
The members of this subsequence converge to the value a = 1 .

Since there are subsequences that converge to different meanings, then the original sequence itself does not converge to any number.

Sequence containing all rational numbers

Now we construct a sequence that contains all rational numbers. Moreover, each rational number will be included in such a sequence an infinite number of times.

The rational number r can be represented as follows:
,
where is an integer; - natural.
We need to assign to each natural number n a pair of numbers p and q so that any pair of p and q is included in our sequence.

To do this, draw axes p and q on the plane. We draw grid lines through integer values p and q . Then each node of this grid with will correspond to a rational number. The whole 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 a single node. This is easy to do if we number the nodes according to the squares whose centers 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. Therefore, they are not shown in the figure.

So, for the upper side of the first square we have:
.
Next, we number the upper part of the next square:

.
We number the upper part of the next square:

.
And so on.

In this way we get a sequence containing all rational numbers. It can be seen 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 elements of which are equal to a predetermined rational number. Since the sequence we have 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 numerical sequence. We also touched upon the issue of its convergence, based on intuitive ideas. Precise definition Convergence is discussed on the Determining the Limit of a Sequence page. Related properties and theorems are outlined on the page

Suppose that each natural number corresponds to a certain real number: the number 1 corresponds to a 1, the number 2 - a 2, the number n - a n. In this case, we say that a numerical sequence is given, which is written as follows: a 1, a 2, ..., and n, where a 1 is the first member, and 2 is the second member, ..., and n is nth term sequences.

There are three main ways to specify a sequence.

1. Analytical. The sequence is given by the formula of the nth term; for example, the formula a n \u003d n / (n + 1) specifies a sequence a 1, a 2, ..., and n, in which

and 1 = 1/(1+1) = 1/2; and 2 \u003d 2 / (2 + 1) \u003d 2/3 ...;

those. sequence 1/2, 2/3, 3/4, …, n/(n + 1).

2. Recurrent. Any member of the sequence is expressed in terms of the preceding members. At this method when specifying a sequence, the first member of the sequence and a formula must be indicated that allows you to calculate any member of the sequence from known previous members.

Let's find several members of the sequence a 1 = 1, a 2 = 1…, and n +2 = a n + a n +1.

a 3 \u003d a 1 + a 2 \u003d 1 + 1 \u003d 2;

a 4 \u003d a 2 + a 3 \u003d 1 + 2 \u003d 3, etc.

As a result, we get the sequence: 1, 1, 2, 3, 5 ....

3. Verbal. This is a sequence assignment by description. For example, a sequence of decimal approximations by the lack of the number e.

Sequences are either ascending or descending.

A sequence (a n), each term of which is less than the one following it, i.e. if a n< а n +1 для любого n, называется возрастающей последовательностью.

A sequence (a n) in which each term is greater than the one following it, i.e. if a n > a n + 1 for any n, is called a descending sequence.

For example:

a) 1, 4, 9, 16, 25, …, n 2 , … – increasing sequence;

b) -1, -2, -3, -4, ..., -n, ... - decreasing sequence;

c) -1, 2, -3, 4, -5, 6, …, (-1) n ∙ n, … is a non-increasing and non-decreasing sequence;

d) 3, 3, 3, 3, 3, 3, …, 3, … is a constant (stationary) sequence.

If each member of the sequence (a n), starting from the second, is equal to the previous one, added with the same number d, then such a sequence is called an arithmetic progression. The number d is called the progression difference.

Thus, the arithmetic progression is given by the equality: a n +1 = a n + d. For example,

a 5 = a 4 + d.

For d > 0, the arithmetic progression increases, for d< 0 убывает.

The sequence 3, 5, 7, 9, 11, 13 ... is an arithmetic progression,
where a 1 \u003d 3, d \u003d 2 (5 - 3, 7 - 5, 9 - 7, etc.).

Sometimes, not the entire sequence, which is an arithmetic progression, is considered, but only its first few members. In this case, one speaks of a finite arithmetic progression.

An arithmetic progression has three properties.

1. Formula of the n-th member of an arithmetic progression:

a n \u003d a 1 + d (n - 1)

2. Formulas for the sum of the first n terms of an arithmetic progression:

a) S n = ((a 1 + a n)/2) ∙ n;

b) S n = ((2a 1 + d(n – 1))/2) ∙ n.

Here S 1 \u003d a 1, S n \u003d a 1 + a 2 + a 3 + ... + a n.

3. Characteristic property of an arithmetic progression: a sequence is an arithmetic sequence if and only if each of its terms, except for the first (and the last in the case of a finite arithmetic progression), is equal to the arithmetic mean of the previous and subsequent terms:

a n \u003d (a n -1 + a n +1) / 2.

If the first term of the sequence (b n) is non-zero and each term, starting from the second, is equal to the previous one multiplied by the same non-zero number q, then such a sequence is called a geometric progression. The number q is called the denominator of the progression.

Thus, the geometric progression is given by the equality b n +1 = b n ∙ q . For example, b 7 = b 6 ∙ q.

The sequence 100, 30, 9, 27/10, ... is a geometric progression, where b 1 = 100, q = 3/10.

A geometric progression is characterized by three properties

1. Formula of the n-th member of a geometric progression:

b n \u003d b 1 ∙ q n -1.

2. Formulas for the sum of the first n members of a geometric progression:

a) S n \u003d (b n q - b 1) / (q - 1);

b) S n = (b 1 (q n - 1)) / (q - 1).

3. Characteristic property of a geometric progression: a sequence is a geometric sequence if and only if each of its members, except for the first (and the last in the case of a finite geometric progression), is associated with the previous and subsequent members by the formula:

b n 2 \u003d b n -1 ∙ b n +1.

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The numerical sequence and its limit are one of the most important problems of mathematics throughout the history of the existence of this science. Constantly updated knowledge, formulated new theorems and proofs - all this allows us to consider this concept from new positions and under different

A numerical sequence, in accordance with one of the most common definitions, is a mathematical function, the basis of which is the set of natural numbers arranged according to one pattern or another.

There are several options for creating number sequences.

Firstly, this function can be specified in the so-called "explicit" way, when there is a certain formula by which each of its members can be determined by simply substituting the ordinal number into the given sequence.

The second method is called "recursive". Its essence lies in the fact that the first few members of the numerical sequence are given, as well as a special recursive formula, with the help of which, knowing the previous member, you can find the next one.

Finally, the most general way of specifying sequences is the so-called when, without much difficulty, one can not only identify one or another term under a certain serial number, but also, knowing several consecutive terms, come to the general formula of this function.

The numerical sequence can be decreasing or increasing. In the first case, each subsequent term is less than the previous one, and in the second, on the contrary, it is greater.

Considering this topic, it is impossible not to touch on the issue of the limits of sequences. The limit of a sequence is such a number when for any quantity, including for an infinitesimal quantity, there is an ordinal number after which the deviation of successive members of the sequence from given point in numerical form becomes less than the value specified during the formation of this function.

The concept of the limit of a numerical sequence is actively used when carrying out certain integral and differential calculations.

Mathematical sequences have a whole set of rather interesting properties.

Firstly, any numerical sequence is an example of a mathematical function, therefore, those properties that are characteristic of functions can be safely applied to sequences. The most striking example of such properties is the provision on increasing and decreasing arithmetic series, which are combined by one general concept are monotonic sequences.

Secondly, there is a fairly large group of sequences that cannot be classified as either increasing or decreasing - these are periodic sequences. In mathematics, they are considered to be those functions in which there is a so-called period length, that is, from a certain moment (n), the following equality begins to operate y n \u003d y n + T, where T will be the very length of the period.