Numeric 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 by some number a n, following a certain law, we get a new series of numbers:

a 1 , a 2 , a 3, , a n –1 , a n , ,

shortly designated and called numerical sequence... The quantity a n is called the common member of the numerical sequence. Usually a numerical sequence is given by some formula a n = f(n) allowing 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 using a common term formula; sometimes a sequence is given by describing its members.

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

A numerical 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, that is, a function of the form f : N  R.

Subsequence
called increasing(diminishing) if for any n  N
Such sequences are called strictly monotone.

Sometimes it is convenient to use not all natural numbers as 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, 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, by specifying a sequence, indicate which 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 . Number sequence 1, 2, 3, 4, 5, ... is a series of natural numbers and has a common term a n = n.

Example 2 . The numerical 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,… - a numerical sequence of approximate values with increasing accuracy.

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

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

Test 6 . The common term in the sequence 1, 2, 6, 24, 120,  is:

Test 7 .
is an:

Test 8 . Common member of the sequence
is an:

Number Sequence Limit

Consider a numerical sequence, the common term of which approaches a certain number A when increasing the serial number n... In this case, the number sequence is said to have a limit. This concept has a stricter definition.

Number A is called the limit of a numerical sequence
:

(1)

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

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

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

Example 5 . Harmonic sequence has a limit of 0. Indeed, for any interval (–; + ) as a number N 0 you can take any integer, more. Then for everyone n > n 0> we have

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

The sequence is called limited if there is such a number M, what
for all n... Any converging sequence is bounded. Any monotonic and bounded sequence has a limit. Any converging sequence has a unique limit.

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

Number e called Euler's number and 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 an:

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 a sequence given by a common term
is equal.

The definition of a numerical sequence is given. Examples of unboundedly increasing, converging and diverging sequences are considered. A sequence containing all rational numbers is considered.

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

A sequence is denoted as the nth member, 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 members. Here are some examples of sequences:
, , .

In other words, a numerical sequence is a function whose domain of definition is the set of natural numbers. The number of elements in a sequence is infinite. Among the elements, there may also be members that have the same values. Also, a sequence can be viewed as a numbered set of numbers, consisting of an infinite number of members.

We will be mainly interested in the question - how the sequences behave when n tends to infinity:. This material is presented in the section Sequence Limit - Basic Theorems and Properties. Here are some examples of sequences.

Sequence examples

Examples of unboundedly increasing sequences

Consider the sequence. Common member of this sequence. Let's write out the first few terms:
.
It can be seen that as the number n grows, the elements grow 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:
.
With the growth of the number n, the elements of this sequence increase indefinitely in absolute value, but do not have a 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 are as follows:
.
It can be seen that as the number n grows, 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 assume that there is an approximate value for the number a = 0 with an error. It is clear that with increasing n this error tends to zero, that is, by choosing n, the error can be made arbitrarily small. 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 limiting value a does not exceed the error ε:.

Next, let's look at the sequence. Her common member. Here are some of its first members:
.
In this sequence, the even numbered terms are zero. Members with odd n are equal. Therefore, with increasing n, their values approach the limiting value a = 0 ... This also follows from the fact that
.
As in the previous example, we can indicate 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 limiting 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 even-numbered members:
,
converge to the value a 1 = 0 ... Odd numbered members:
,
converge to the value a 2 = 2 ... The sequence itself, with increasing n, does not converge to any value.

Sequence with members distributed in the interval (0; 1)

Now let's look at a more interesting sequence. Take a segment on the number line. Let's divide it in half. We get two line segments. Let be
.
We divide each of the segments in half again. We get four line segments. Let be
.
Divide each segment in half again. Let's take

.
Etc.

As a result, we get a sequence, the elements of which are distributed in the open interval (0; 1) ... Whichever 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 select a subsequence that converges 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 previously selected point.

For example, for point a = 0 the following subsequence can be selected:
.
= 0 .

For point a = 1 choose the following subsequence:
.
The members 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 build a sequence that contains all the rational numbers. Moreover, each rational number will appear in such a sequence an infinite number of times.

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

To do this, draw the p and q axes on the plane. Draw the grid lines through the integer values of p and q. Then each node of this grid 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 you number the nodes by squares whose centers are located at the point (0; 0) (see figure). Moreover, 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 the top of the next square:

.
We number the top of the next square:

.
Etc.

In this way, we get a sequence containing all rational numbers. You can see that any rational number appears in this sequence an infinite number of times. Indeed, along with a 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 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 a number sequence. We also touched on the issue of its convergence, based on intuitive ideas. Precise definition Convergence is discussed on the Sequence Limit Definition page. Related properties and theorems are outlined on the page

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

There are three main ways of sequencing.

1. Analytical. The sequence is given by the formula of the nth term; for example, the formula а n = n / (n + 1) defines the sequence а 1, а 2, ..., а n, in which

a 1 = 1 / (1 + 1) = 1/2; a 2 = 2 / (2 + 1) = 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 way specifying a sequence, the first member of the sequence and a formula that allows you to calculate any member of the sequence by the known previous members must be indicated.

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

a 3 = a 1 + a 2 = 1 + 1 = 2;

a 4 = a 2 + a 3 = 1 + 2 = 3, etc.

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

3. Verbal. This is the sequencing of the description. For example, a sequence of decimal approximations for the lack of e.

Sequences are ascending and descending.

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

Sequence (a n), each member of which is greater than the next one, i.e. if a n> a n +1 for any n, is called a decreasing 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,… - 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 difference in progression.

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 = 3, d = 2 (5 - 3, 7 - 5, 9 - 7, etc.).

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

Arithmetic progression has three properties.

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

a n = 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 = a 1, S n = a 1 + a 2 + a 3 +… + a n.

3. The characteristic property of the arithmetic progression: a sequence is an arithmetic sequence if and only if each of its members, 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 members:

a n = (a n -1 + a n +1) / 2.

If the first term of the sequence (b n) is nonzero and each term starting from the second is equal to the previous one multiplied by the same nonzero 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.

The geometric progression is characterized by three properties

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

b n = b 1 ∙ q n -1.

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

a) S n = (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 = b n -1 ∙ b n +1.

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The numerical sequence and its limit represent 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 based on a set of natural numbers arranged according to one or another pattern.

There are several options for creating number sequences.

First, this function can be specified in the so-called "explicit" way, when there is a certain formula, with the help of which each of its members can be determined by simple substitution of the ordinal number in the given sequence.

The second method is called "recurrent". Its essence lies in the fact that the first few members of the numerical sequence are set, as well as a special recursive formula, with the help of which, knowing the previous term, 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 ordinal number, but also, knowing several consecutive terms, come to a general formula for a given function.

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

Considering this topic, one cannot but touch upon the question of the limits of sequences. The limit of a sequence is a number such that for any, including an infinitesimal quantity, there is a sequence number, after which the deviation of successive members of the sequence from set point in numerical form, it becomes less than the value specified when this function was formed.

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

Mathematical sequences have a whole set of quite interesting properties.

First, 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 united by one general concept- monotonic sequences.

Secondly, there is a fairly large group of sequences that cannot be attributed to either ascending or descending - these are periodic sequences. In mathematics, they are considered to be those functions in which the so-called period length exists, that is, from a certain moment (n), the following equality y n = y n + T begins to operate, where T will be the same period length.