Number sequence.
First, let's think about the word itself: what is sequence? Sequence is when something follows something. For example, a sequence of actions, a sequence of seasons. Or when someone is located behind someone. For example, a sequence of people in a queue, a sequence of elephants on the path to a watering hole.
Let's clarify immediately characteristic features sequences. Firstly, sequence members are located strictly in in a certain order . So, if two people in the queue are swapped, then this will already be other subsequence. Secondly, everyone sequence member You can assign a serial number:
It's the same with numbers. Let to each natural value according to some rule compliant real number. Then they say that a numerical sequence is given.
Yes, in mathematical problems Unlike life situations the sequence almost always contains infinitely many numbers.
Wherein:
Called first member sequences;
– second member sequences;
– third member sequences;
– nth or common member sequences;
In practice, the sequence is usually given common term formula, For example:
– sequence of positive even numbers:
Thus, the record uniquely determines all members of the sequence - this is the rule (formula) by which natural values 
numbers are put into correspondence. Therefore, the sequence is often briefly denoted by a common term, and instead of “x” others can be used letters, For example:
Sequence of positive odd numbers:
Another common sequence:
As many have probably noticed, the “en” variable plays the role of a kind of counter.
In fact, we dealt with number sequences back in middle school. Let's remember arithmetic progression. I won’t rewrite the definition; let’s touch on the essence with a specific example. Let be the first term, and – step arithmetic progression. Then:
– the second term of this progression;
– the third term of this progression;
- fourth;
- fifth;
And, obviously, the nth term is given recurrent formula
Note: V recurrent formula each subsequent member is expressed through the previous member or even through a whole set of previous members.
The resulting formula is of little use in practice - to get, say, to , you need to go through all the previous terms. And in mathematics, a more convenient expression for the nth term of an arithmetic progression has been derived: 
. In our case:
Substitute natural numbers into the formula and check the correctness of the one constructed above number sequence.
Similar calculations can be made for geometric progression, the nth term of which is given by the formula , where is the first term, and – denominator progression. In math tasks, the first term is often equal to one.
Examples:
progression sets the sequence 
;
progression 
sets the sequence;
progression 
sets the sequence 
;
progression 
sets the sequence 
.
I hope everyone knows that –1 to an odd power is equal to –1, and to an even power – one.
Progression is called infinitely decreasing, if (last two cases).
Let's add two new friends to our list, one of whom has just knocked on the monitor's matrix:
The sequence in mathematical jargon is called a “blinker”:
Thus, sequence members can be repeated. So, in the example considered, the sequence consists of two infinitely alternating numbers.
Does it happen that the sequence consists of identical numbers? Certainly. For example, it asks infinite number"threes". For aesthetes, there is a case when “en” still formally appears in the formula:
Factorial:
Just a condensed recording of the work:
Not a graphomania at all, it will be useful for tasks;-) I recommend understanding it, remembering it and even copying it into a notebook. ...One question came to mind: why doesn’t anyone create such useful graffiti? A man is riding on a train, looking out the window and studying factorials. Punks are resting =)
Perhaps some readers still do not fully understand how to describe the members of a sequence, knowing the common member. That rare case, when the control shot returns to life:
Let's deal with the sequence 
.
First, let’s substitute the value into the nth term and carefully carry out the calculations:
Then we plug in the following number:
Four:
Well, now there’s no shame in earning an excellent mark:
The concept of a sequence limit.
To better understand the following information, it is advisable to UNDERSTAND what it is limit of a function. Of course, in the standard course mathematical analysis first they consider the limit of the sequence and only then the limit of the function, but the fact is that I have already talked in detail about the very essence of the limit. Moreover, in theory, a number sequence is considered a special case of a function, and people who are familiar with the limit of a function will have much more fun.
Let's invite a simple friend to dance:
What happens when "en" increases to infinity? Obviously, the members of the sequence will be infinitely close approach zero. This is the limit of this sequence, which is written as follows:
If the limit of the sequence equal to zero, then it is called infinitesimal.
In the theory of mathematical analysis it is given strict definition of the sequence limit through the so-called epsilon neighborhood. The next article will be devoted to this definition, but for now let’s look at its meaning:
Let us depict on the number line the terms of the sequence and the neighborhood symmetric with respect to zero (limit):
Now pinch the blue area with the edges of your palms and begin to reduce it, pulling it towards the limit (red point). A number is the limit of a sequence if FOR ANY pre-selected -neighborhood (as small as you like) will be inside it infinitely many members of the sequence, and OUTSIDE it - only final number of members (or none at all). That is, the epsilon neighborhood can be microscopic, and even smaller, but the “infinite tail” of the sequence must sooner or later completely enter this neighborhood.
There is even such a task - prove the limit of the sequence using the definition.
The sequence is also infinitesimal: with the difference that its members do not jump back and forth, but approach the limit exclusively from the right.
Naturally, the limit can be equal to any other finite number, elementary example:
Here the fraction tends to zero, and accordingly, the limit is equal to “two”.
If the sequence exists final limit , then it is called convergent(in particular, infinitesimal at ). IN otherwise – divergent, in this case, two options are possible: either the limit does not exist at all, or it is infinite. IN the latter case the sequence is called infinitely large. Let's gallop through the examples of the first paragraph:
Sequences 
are infinitely large, as their members confidently move towards “plus infinity”:
An arithmetic progression with the first term and step is also infinitely large:
By the way, any arithmetic progression diverges, with the exception of the case with zero step - when to specific number is added endlessly. The limit of such a sequence exists and coincides with the first term.
The sequences have a similar fate:
Any infinitely decreasing geometric progression, as is clear from the name, infinitely small:
If the denominator of the geometric progression is , then the sequence is infinitely large:
If, for example, then the limit does not exist at all, since the members tirelessly jump either to “plus infinity” or to “minus infinity”. A common sense and Matan’s theorems suggest that if something strives somewhere, then this is the only cherished place.
After a little revelation 
it becomes clear that the “flashing light” is to blame for the uncontrollable throwing, which, by the way, diverges on its own.
Indeed, for a sequence it is easy to choose a -neighborhood that, say, only clamps the number –1. As a result, an infinite number of sequence members (“plus ones”) will remain outside this neighborhood. But by definition, the “infinite tail” of the sequence from a certain moment (natural number) must fully go into ANY vicinity of your limit. Conclusion: the sky is the limit.
Factorial is infinitely large sequence:
Moreover, it is growing by leaps and bounds, so it is a number that has more than 100 digits (digits)! Why exactly 70? On it my engineering microcalculator begs for mercy.
With a control shot, everything is a little more complicated, and we have just come to the practical part of the lecture, in which we will analyze combat examples:
How to find the limit of a sequence.
But now it is necessary to be able to solve the limits of functions, at least at the level of two basic lessons: Limits. Examples of solutions And Wonderful Limits. Because many solution methods will be similar. But, first of all, let’s analyze the fundamental differences between the limit of a sequence and the limit of a function:
In the limit of the sequence, the “dynamic” variable “en” can tend to only to “plus infinity”– towards increasing natural numbers 
.
In the limit of the function, “x” can be directed anywhere – to “plus/minus infinity” or to an arbitrary real number.
Subsequence discrete(discontinuous), that is, it consists of individual isolated members. One, two, three, four, five, the bunny went out for a walk. The argument of a function is characterized by continuity, that is, “X” smoothly, without incident, tends to one or another value. And, accordingly, the function values will also continuously approach their limit.
Because of discreteness within the sequences there are their own signature things, such as factorials, “flashing lights”, progressions, etc. And now I will try to analyze the limits that are specific to sequences.
Let's start with progressions:
Example 1
Solution: something similar to an infinitely decreasing geometric progression, but is it really that? For clarity, let’s write down the first few terms: 
Since, then we are talking about amount terms of an infinitely decreasing geometric progression, which is calculated by the formula.
Let's make a decision:
We use the formula for the sum of an infinitely decreasing geometric progression: . IN in this case: – the first term, – the denominator of the progression.
The main thing is to cope with four-story fraction:
Eat.
Example 2
Write the first four terms of the sequence and find its limit 
This is an example for independent decision. To eliminate the uncertainty in the numerator, you will need to apply the formula for the sum of the first terms of an arithmetic progression:
, where is the first and a is the nth term of the progression.
Since within sequences "en" always tends to "plus infinity", it is not surprising that uncertainty is one of the most popular.
And many examples are solved in exactly the same way as function limits!
How to calculate these limits? See Examples No. 1-3 of the lesson Limits. Examples of solutions.
Or maybe something more complicated like 
? Check out Example No. 3 of the article Methods for solving limits.
From a formal point of view, the difference will be only in one letter - “x” here, and “en” here.
The technique is the same - the numerator and denominator must be divided by “en” to the highest degree.
Also, uncertainty within sequences is quite common. How to solve limits like 
can be found in Examples No. 11-13 of the same article.
To understand the limit, refer to Example No. 7 of the lesson Wonderful Limits(second wonderful limit is also valid for the discrete case). The solution will again be like a carbon copy with a single letter difference.
The next four examples (Nos. 3-6) are also “two-faced”, but in practice for some reason they are more characteristic of sequence limits than of function limits:
Example 3
Find the limit of the sequence
Solution: at first complete solution, then step-by-step comments:
(1) In the numerator we use the formula twice.
(2) We present similar terms in the numerator.
(3) To eliminate uncertainty, divide the numerator and denominator by (“en” to the highest degree).
As you can see, nothing complicated.
Example 4
Find the limit of the sequence
This is an example for you to solve on your own, abbreviated multiplication formulas to help.
Within s indicative Sequences use a similar method of dividing the numerator and denominator:
Example 5
Find the limit of the sequence
Solution Let's arrange it according to the same scheme:
(1) Using properties of degrees, let’s remove everything unnecessary from the indicators, leaving only “en” there.
(2) We look at what exponential sequences are in the limit: and choose a sequence with the largest basis: . To eliminate uncertainty, divide the numerator and denominator by .
(3) We carry out term-by-term division in the numerator and denominator. Since it is infinitely decreasing geometric progression, then it tends to zero. And even more so, the constant divided by the increasing progression tends to zero: . We make the appropriate notes and write down the answer.
Example 6
Find the limit of the sequence
This is an example for you to solve on your own.
Somehow, undeservedly, stylish handwriting, inherent only to the limit of consistency, remained in oblivion. It's time to fix the situation:
Example 7
Find the limit of the sequence
Solution: to get rid of the “eternal rival” you need to write the factorials in the form of products. But before we start with mathematical graffiti, let's consider specific example, For example: .
The last factor in the product is six. What needs to be done to get the previous multiplier? Subtract one: 6 – 1 = 5. To get a multiplier, which is located even further, you need to subtract one from five again: 5 – 1 = 4. And so on.
Don't worry, this is not a first grade lesson. correctional school, In fact we get acquainted with an important and universal algorithm entitled " how to expand any factorial" Let's deal with the most malicious flooder in our chat:
Obviously, the last factor in the product will be .
How to get the previous multiplier? Subtract one:
How to get great-grandfather? Subtract one again: .
Well, let's move one step deeper:
Thus, our monster will sign as follows:
With numerator factorials everything is simpler, okay, little hooligans.
Let's make a decision:
(1) We describe factorials
(2) The numerator has TWO terms. We take out of brackets everything that can be taken out, in this case this is the work. Square brackets, as I said somewhere a couple of times, differ from parentheses only in their squareness.
(3) Reduce the numerator and denominator by .... ...hmmm, there really is a lot of fluff here.
(4) Simplify the numerator
(5) Reduce the numerator and denominator by . Here in to a certain extent lucky. IN general case at the top and bottom you get ordinary polynomials, after which you have to perform the standard action - divide the numerator and denominator by “en” to the highest power.
More advanced students who can easily break down factorials in their heads can solve the example much faster. At the first step, we divide the numerator by the denominator term by term and mentally perform the abbreviations:
But the decomposition method is still more thorough and reliable.
Example 8
Find the limit of the sequence
As in any society, among the number sequences there are extravagant individuals.
Theorem: work limited sequence to an infinitesimal sequence - there is an infinitesimal sequence.
If you do not really understand the term “limitation”, please study the article about elementary functions and graphs.
A similar theorem is true, by the way, for functions: the product limited function on indefinitely small function- is an infinitesimal function.
Example 9
Find the limit of the sequence
Solution: sequence – limited: 
, and the sequence is infinitely small, which means, according to the corresponding theorem: 
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)
called the law (rule), according to which, everyone 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
Numerical sequence is a numerical function defined on the set of natural numbers .
If the function is defined on the set of natural numbers 
, then the set of function values will be countable and each number 
matches the number 
. In this case they say that it is given number sequence. Numbers are called elements or members of a sequence, and the number 
– general or 
-th member of the sequence. Each element 
has a subsequent element 
. 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 number is calculated 
, i.e. indicating its formula 
‑th member 
.
Example.Subsequence
can be given by the formula:
.
Usually sequences are denoted as follows: etc., where the formula for it is indicated in brackets 
th member.
Example.Subsequence
‑this is a sequence
The set of all elements of a sequence 
denoted by 
.
Let 
And 
- two sequences.
WITH ummah sequences 
And 
called a sequence 
, Where 
, i.e..
R difference of these sequences is called a sequence 
, Where 
, i.e..
If 
And
‑
constants, then the sequence
,
called linear combination
sequences 
And 
, i.e.
The work sequences 
And 
called the sequence with 
th member 
, i.e. 
.
If 
, then we can 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 quotient are equal 
.
If you cross out some elements of the sequence 
so that it remains infinite set elements, then we get another sequence called subsequence sequences 
. If you cross out the first few elements of the sequence 
, That new sequence called the 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 remainders is bounded.
Converging sequences
They say that subsequence 
converges if there is a number 
such that for anyone 
there is such a thing 
that for anyone 
, the inequality holds: 
.
Number 
called limit of the sequence
. At the same time they write down 
or 
.
Example.
.
Let's show that 
. Let's set any number 
. Inequality 
performed for 
, such that 
, that the definition of convergence is carried out 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 
(if) the elements of the sequence are in the interval 
which is called 
–neighborhood of the point 
.
Subsequence 
, whose limit is zero ( 
, or 
at 
) is called infinitesimal.
In relation to infinitesimals, the following statements are true:
The sum of two infinitesimals is infinitesimal;
The product of an infinitesimal and a finite quantity is infinitesimal.
Theorem
.In order for the sequence 
had a limit, it was necessary and sufficient for 
, Where 
– constant; 
– infinitesimal
.
Basic properties of convergent sequences:
Properties 3. and 4. are generalized 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 
, fraction limit equal to the limit relations of senior members (i.e. members containing the greatest degrees 
numerator and denominator).
Subsequence 
called:
All such sequences are called monotonous.
Theorem
.
If the sequence 
monotonically increasing and bounded above, then it converges and its limit is equal to its exact top edge; if the sequence is decreasing and bounded below, then it converges to its infimum.
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 an infinite number of 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 
.
