Mathematics is the science that builds the world. Both the scientist and the common man - no one can do without it. First, young children are taught to count, then add, subtract, multiply and divide, by the middle school, letter designations come into play, and in the older one they can no longer be dispensed with.

But today we will talk about what all known mathematics is based on. About the community of numbers called "sequence limits".

What are sequences and where is their limit?

The meaning of the word "sequence" is not difficult to interpret. This is such a construction of things, where someone or something is located in a certain order or queue. For example, the queue for tickets to the zoo is a sequence. And there can only be one! If, for example, you look at the queue to the store, this is one sequence. And if one person suddenly leaves this queue, then this is a different queue, a different order.

The word "limit" is also easily interpreted - this is the end of something. However, in mathematics, the limits of sequences are those values ​​on the number line that a sequence of numbers tends to. Why strives and does not end? It's simple, the number line has no end, and most sequences, like rays, have only a beginning and look like this:

x 1, x 2, x 3, ... x n ...

Hence the definition of a sequence is a function of the natural argument. In simpler words, it is a series of members of some set.

How is a number sequence built?

The simplest example of a number sequence might look like this: 1, 2, 3, 4, …n…

In most cases, for practical purposes, sequences are built from numbers, and each next member of the series, let's denote it by X, has its own name. For example:

x 1 - the first member of the sequence;

x 2 - the second member of the sequence;

x 3 - the third member;

x n is the nth member.

In practical methods, the sequence is given by a general formula in which there is some variable. For example:

X n \u003d 3n, then the series of numbers itself will look like this:

It is worth remembering that in the general notation of sequences, you can use any Latin letters, and not just X. For example: y, z, k, etc.

Arithmetic progression as part of sequences

Before looking for the limits of sequences, it is advisable to delve deeper into the very concept of such a number series, which everyone encountered when they were in the middle classes. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.

Task: “Let a 1 \u003d 15, and the step of the progression of the number series d \u003d 4. Build the first 4 members of this row"

Solution: a 1 = 15 (by condition) is the first member of the progression (number series).

and 2 = 15+4=19 is the second member of the progression.

and 3 \u003d 19 + 4 \u003d 23 is the third term.

and 4 \u003d 23 + 4 \u003d 27 is the fourth term.

However, with this method it is difficult to reach large values, for example, up to a 125. . Especially for such cases, a formula convenient for practice was derived: a n \u003d a 1 + d (n-1). In this case, a 125 \u003d 15 + 4 (125-1) \u003d 511.

Sequence types

Most of the sequences are endless, it's worth remembering for a lifetime. There are two interesting types of number series. The first is given by the formula a n =(-1) n . Mathematicians often refer to this flasher sequences. Why? Let's check its numbers.

1, 1, -1 , 1, -1, 1, etc. With this example, it becomes clear that numbers in sequences can easily be repeated.

factorial sequence. It is easy to guess that there is a factorial in the formula that defines the sequence. For example: and n = (n+1)!

Then the sequence will look like this:

and 2 \u003d 1x2x3 \u003d 6;

and 3 \u003d 1x2x3x4 \u003d 24, etc.

A sequence given by an arithmetic progression is called infinitely decreasing if the inequality -1 is observed for all its members

and 3 \u003d - 1/8, etc.

There is even a sequence consisting of the same number. So, and n \u003d 6 consists of an infinite number of sixes.

Determining the Limit of a Sequence

Sequence limits have long existed in mathematics. Of course, they deserve their own competent design. So, time to learn the definition of sequence limits. First, consider the limit for a linear function in detail:

1. All limits are abbreviated as lim.
2. The limit entry consists of the abbreviation lim, some variable tending to a certain number, zero or infinity, as well as the function itself.

It is easy to understand that the definition of the limit of a sequence can be formulated as follows: it is a certain number, to which all members of the sequence infinitely approach. Simple example: and x = 4x+1. Then the sequence itself will look like this.

5, 9, 13, 17, 21…x…

Thus, this sequence will increase indefinitely, which means that its limit is equal to infinity as x→∞, and this should be written as follows:

If we take a similar sequence, but x tends to 1, we get:

And the series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number more and more close to one (0.1, 0.2, 0.9, 0.986). It can be seen from this series that the limit of the function is five.

From this part, it is worth remembering what the limit of a numerical sequence is, the definition and method for solving simple tasks.

General notation for the limit of sequences

Having analyzed the limit of the numerical sequence, its definition and examples, we can proceed to a more complex topic. Absolutely all limits of sequences can be formulated by one formula, which is usually analyzed in the first semester.

So, what does this set of letters, modules and inequality signs mean?

∀ is a universal quantifier, replacing the phrases “for all”, “for everything”, etc.

∃ is an existence quantifier, in this case it means that there is some value N belonging to the set of natural numbers.

A long vertical stick following N means that the given set N is "such that". In practice, it can mean "such that", "such that", etc.

To consolidate the material, read the formula aloud.

Uncertainty and certainty of the limit

The method of finding the limit of sequences, which was discussed above, although simple to use, is not so rational in practice. Try to find the limit for this function:

If we substitute different x values ​​(increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. It turns out a rather strange fraction:

But is it really so? Calculating the limit of the numerical sequence in this case seems easy enough. It would be possible to leave everything as it is, because the answer is ready, and it was received on reasonable terms, but there is another way specifically for such cases.

First, let's find the highest degree in the numerator of the fraction - this is 1, since x can be represented as x 1.

Now let's find the highest degree in the denominator. Also 1.

Divide both the numerator and the denominator by the variable to the highest degree. In this case, we divide the fraction by x 1.

Next, let's find what value each term containing the variable tends to. In this case, fractions are considered. As x→∞, the value of each of the fractions tends to zero. When making a paper in writing, it is worth making the following footnotes:

The following expression is obtained:

Of course, the fractions containing x did not become zeros! But their value is so small that it is quite permissible not to take it into account in the calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero.

What is a neighborhood?

Let us assume that the professor has at his disposal a complex sequence, given, obviously, by a no less complex formula. The professor found the answer, but does it fit? After all, all people make mistakes.

Auguste Cauchy came up with a great way to prove the limits of sequences. His method was called neighborhood operation.

Suppose that there is some point a, its neighborhood in both directions on the real line is equal to ε ("epsilon"). Since the last variable is distance, its value is always positive.

Now let's set some sequence x n and suppose that the tenth member of the sequence (x 10) is included in the neighborhood of a. How to write this fact in mathematical language?

Suppose x 10 is to the right of point a, then the distance x 10 -a<ε, однако, если расположить «икс десятое» левее точки а, то расстояние получится отрицательным, а это невозможно, значит, следует занести левую часть неравенства под модуль. Получится |х 10 -а|<ε.

Now it is time to explain in practice the formula mentioned above. It is fair to call some number a the end point of a sequence if the inequality ε>0 holds for any of its limits, and the entire neighborhood has its own natural number N, such that all members of the sequence with higher numbers will be inside the sequence |x n - a|< ε.

With such knowledge, it is easy to solve the limits of a sequence, to prove or disprove a ready answer.

Theorems

Theorems on the limits of sequences are an important component of the theory, without which practice is impossible. There are only four main theorems, remembering which, you can significantly facilitate the process of solving or proving:

1. Uniqueness of the limit of a sequence. Any sequence can have only one limit or not at all. The same example with a queue that can only have one end.
2. If a series of numbers has a limit, then the sequence of these numbers is limited.
3. The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
4. The quotient limit of two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.

Sequence Proof

Sometimes it is required to solve an inverse problem, to prove a given limit of a numerical sequence. Let's look at an example.

Prove that the limit of the sequence given by the formula is equal to zero.

According to the above rule, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим:

Let's express n in terms of "epsilon" to show the existence of a certain number and prove the existence of a sequence limit.

At this stage, it is important to recall that "epsilon" and "en" are positive numbers and not equal to zero. Now you can continue further transformations using the knowledge about inequalities gained in high school.

Whence it turns out that n > -3 + 1/ε. Since it is worth remembering that we are talking about natural numbers, the result can be rounded by putting it in square brackets. Thus, it was proved that for any value of the “epsilon” neighborhood of the point a = 0, a value was found such that the initial inequality is satisfied. From this we can safely assert that the number a is the limit of the given sequence. Q.E.D.

With such a convenient method, you can prove the limit of a numerical sequence, no matter how complicated it may seem at first glance. The main thing is not to panic at the sight of the task.

Or maybe he doesn't exist?

The existence of a sequence limit is not necessary in practice. It is easy to find such series of numbers that really have no end. For example, the same flasher x n = (-1) n . it is obvious that a sequence consisting of only two digits cyclically repeating cannot have a limit.

The same story is repeated with sequences consisting of a single number, fractional, having in the course of calculations an uncertainty of any order (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculation also takes place. Sometimes rechecking your own solution will help you find the limit of successions.

monotonic sequence

Above, we considered several examples of sequences, methods for solving them, and now let's try to take a more specific case and call it a "monotone sequence".

Definition: it is fair to call any sequence monotonically increasing if it satisfies the strict inequality x n< x n +1. Также любую последовательность справедливо называть монотонной убывающей, если для неё выполняется неравенство x n >x n +1.

Along with these two conditions, there are also similar non-strict inequalities. Accordingly, x n ≤ x n +1 (non-decreasing sequence) and x n ≥ x n +1 (non-increasing sequence).

But it is easier to understand this with examples.

The sequence given by the formula x n \u003d 2 + n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.

And if we take x n \u003d 1 / n, then we get a series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing sequence.

Limit of convergent and bounded sequence

A bounded sequence is a sequence that has a limit. A convergent sequence is a series of numbers that has an infinitesimal limit.

Thus, the limit of a bounded sequence is any real or complex number. Remember that there can only be one limit.

The limit of a convergent sequence is an infinitesimal quantity (real or complex). If you draw a sequence diagram, then at a certain point it will, as it were, converge, tend to turn into a certain value. Hence the name - convergent sequence.

Monotonic sequence limit

Such a sequence may or may not have a limit. First, it is useful to understand when it is, from here you can start when proving the absence of a limit.

Among monotonic sequences, convergent and divergent are distinguished. Convergent - this is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent - a sequence that has no limit in its set (neither real nor complex).

Moreover, the sequence converges if its upper and lower limits converge in a geometric representation.

The limit of a convergent sequence can in many cases be equal to zero, since any infinitesimal sequence has a known limit (zero).

Whichever convergent sequence you take, they are all bounded, but far from all bounded sequences converge.

The sum, difference, product of two convergent sequences is also a convergent sequence. However, the quotient can also converge if it is defined!

Various actions with limits

Sequence limits are as significant (in most cases) as numbers and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits.

First, just like digits and numbers, the limits of any sequence can be added and subtracted. Based on the third theorem on the limits of sequences, the following equality is true: the limit of the sum of sequences is equal to the sum of their limits.

Secondly, based on the fourth theorem on the limits of sequences, the following equality is true: the limit of the product of the nth number of sequences is equal to the product of their limits. The same is true for division: the limit of the quotient of two sequences is equal to the quotient of their limits, provided that the limit is not equal to zero. After all, if the limit of sequences is equal to zero, then division by zero will turn out, which is impossible.

Sequence Value Properties

It would seem that the limit of the numerical sequence has already been analyzed in some detail, but such phrases as “infinitely small” and “infinitely large” numbers are mentioned more than once. Obviously, if there is a sequence 1/x, where x→∞, then such a fraction is infinitely small, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such values ​​have their own characteristics. The properties of the limit of a sequence having arbitrary small or large values ​​are as follows:

1. The sum of any number of arbitrarily small quantities will also be a small quantity.
2. The sum of any number of large values ​​will be an infinitely large value.
3. The product of arbitrarily small quantities is infinitely small.
4. The product of arbitrarily large numbers is an infinitely large quantity.
5. If the original sequence tends to an infinite number, then the reciprocal of it will be infinitesimal and tend to zero.

In fact, calculating the limit of a sequence is not such a difficult task if you know a simple algorithm. But the limits of sequences are a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution of such expressions. Starting small, over time, you can reach big heights.

Statements of the main theorems and properties of numerical sequences with limits are given. Contains the definition of a sequence and its limit. Arithmetic operations with sequences, properties related to inequalities, convergence criteria, properties of infinitely small and infinitely large sequences are considered.

Content

Properties of finite limits of sequences

Basic properties

A point a is the limit of a sequence if and only if outside any neighborhood of this point is finite number of elements sequences or the empty set.

If the number a is not the limit of the sequence , then there is such a neighborhood of the point a , outside of which there is infinite number of sequence elements.

Uniqueness theorem for the limit of a number sequence. If a sequence has a limit, then it is unique.

If a sequence has a finite limit, then it limited.

If each element of the sequence is equal to the same number C : , then this sequence has a limit equal to the number C .

If the sequence add, drop or change the first m elements, then this will not affect its convergence.

Proofs of basic properties given on the page
Basic properties of finite limits of sequences >>> .

Arithmetic with limits

Let there be finite limits and sequences and . And let C be a constant, that is, a given number. Then
;
;
;
, if .
In the case of the quotient, it is assumed that for all n .

If , then .

Arithmetic property proofs given on the page
Arithmetic properties of finite limits of sequences >>> .

Properties associated with inequalities

If the elements of the sequence, starting from some number, satisfy the inequality , then the limit a of this sequence also satisfies the inequality .

If the elements of the sequence, starting from a certain number, belong to a closed interval (segment) , then the limit a also belongs to this interval: .

If and and elements of sequences, starting from some number, satisfy the inequality , then .

If and, starting from some number, , then .
In particular, if, starting from some number, , then
if , then ;
if , then .

If and , then .

Let and . If a < b , then there is a natural number N such that for all n > N the inequality is satisfied.

Proofs of properties related to inequalities given on the page
Properties of sequence limits related to >>> inequalities.

Infinitesimal and infinitesimal sequences

Infinitesimal sequence

An infinitesimal sequence is a sequence whose limit is zero:
.

Sum and Difference finite number of infinitesimal sequences is an infinitesimal sequence.

Product of a bounded sequence to an infinitesimal is an infinitesimal sequence.

Product of a finite number infinitesimal sequences is an infinitesimal sequence.

For a sequence to have a limit a , it is necessary and sufficient that , where is an infinitesimal sequence.

Proofs of properties of infinitesimal sequences given on the page
Infinitely small sequences - definition and properties >>> .

Infinitely large sequence

An infinitely large sequence is a sequence that has an infinitely large limit. That is, if for any positive number there is such a natural number N , depending on , that for all natural numbers the inequality
.
In this case, write
.
Or at .
They say it tends to infinity.

If , starting from some number N , then
.
If , then
.

If the sequences are infinitely large, then starting from some number N , a sequence is defined that is infinitely small. If are an infinitesimal sequence with non-zero elements, then the sequence is infinitely large.

If the sequence is infinitely large and the sequence is bounded, then
.

If the absolute values ​​of the elements of the sequence are bounded from below by a positive number (), and is infinitely small with non-zero elements, then
.

In details definition of an infinitely large sequence with examples given on the page
Definition of an infinitely large sequence >>> .
Proofs for properties of infinitely large sequences given on the page
Properties of infinitely large sequences >>> .

Sequence Convergence Criteria

Monotonic sequences

A strictly increasing sequence is a sequence for all elements of which the following inequalities hold:
.

Similar inequalities define other monotone sequences.

Strictly decreasing sequence:
.
Non-decreasing sequence:
.
Non-increasing sequence:
.

It follows that a strictly increasing sequence is also nondecreasing. A strictly decreasing sequence is also non-increasing.

A monotonic sequence is a non-decreasing or non-increasing sequence.

A monotonic sequence is bounded on at least one side by . A non-decreasing sequence is bounded from below: . A non-increasing sequence is bounded from above: .

Weierstrass theorem. In order for a non-decreasing (non-increasing) sequence to have a finite limit, it is necessary and sufficient that it be bounded from above (from below). Here M is some number.

Since any non-decreasing (non-increasing) sequence is bounded from below (from above), the Weierstrass theorem can be rephrased as follows:

For a monotone sequence to have a finite limit, it is necessary and sufficient that it be bounded: .

Monotonic unbounded sequence has an infinite limit, equal for non-decreasing and non-increasing sequences.

Proof of the Weierstrass theorem given on the page
Weierstrass' theorem on the limit of a monotone sequence >>> .

Cauchy criterion for sequence convergence

Cauchy condition
Consistency satisfies Cauchy condition, if for any there exists a natural number such that for all natural numbers n and m satisfying the condition , the inequality
.

A fundamental sequence is a sequence that satisfies the Cauchy condition.

Cauchy criterion for sequence convergence. For a sequence to have a finite limit, it is necessary and sufficient that it satisfies the Cauchy condition.

Proof of the Cauchy Convergence Criterion given on the page
Cauchy's convergence criterion for a sequence >>> .

Subsequences

Bolzano-Weierstrass theorem. From any bounded sequence, a convergent subsequence can be distinguished. And from any unlimited sequence - an infinitely large subsequence converging to or to .

Proof of the Bolzano-Weierstrass theorem given on the page
Bolzano–Weierstrass theorem >>> .

Definitions, theorems, and properties of subsequences and partial limits are discussed on page
Subsequences and partial limits of sequences >>>.

References:
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
V.A. Zorich. Mathematical analysis. Part 1. Moscow, 1997.
V.A. Ilyin, E.G. Poznyak. Fundamentals of mathematical analysis. Part 1. Moscow, 2005.

See also:

Xn are elements or sequence members, n is a sequence member. If the function f(n) is given analytically, that is, by a formula, then xn=f(n) is called the formula of a member of the sequence.

The number a is called the limit of the sequence (xn) if for any ε>0 there exists a number n=n(ε) starting from which the inequality |xn-a |

Example 2. Prove that under the conditions of Example 1 the number a=1 is not the limit of the sequence of the previous example. Decision. Simplify the common term of the sequence again. Take ε=1 (this is any number >

The problems of directly calculating the limit of a sequence are rather monotonous. All of them contain ratios of polynomials with respect to n or irrational expressions with respect to these polynomials. When starting to solve, take out the parentheses (radical sign) of the component that is in the highest degree. Suppose that for the numerator of the original expression this will lead to the appearance of the factor a^p, and for the denominator b^q. Obviously, all the remaining terms have the form C / (n-k) and tend to zero when n>

The first way to calculate the limit of a sequence is based on its definition. True, it should be remembered that it does not give ways to directly search for the limit, but only allows you to prove that some number a is (or is not) a limit. Example 1. Prove that the sequence (xn) = ((3n ^ 2-2n -1)/(n^2-n-2)) has a limit a=3. Solution. Prove by applying the definition in reverse order. That is, from right to left. First check if it is possible to simplify the formula for xn.хn =(3n^2+4n+2)/(n^2+3n22)=((3n+1)(n+1))/((n+2) (n+1))=)=(3n+1)/(n+2). Consider the inequality |(3n+1)/(n+2)-3|0 you can find any natural number nε greater than -2+ 5/ε.

Example 2. Prove that under the conditions of Example 1 the number a=1 is not the limit of the sequence of the previous example. Decision. Simplify the common term of the sequence again. Take ε=1 (this is any number >0). Write down the final inequality of the general definition |(3n+1)/(n+2)-1|

The problems of directly calculating the limit of a sequence are rather monotonous. All of them contain ratios of polynomials with respect to n or irrational expressions with respect to these polynomials. When starting to solve, take out the parentheses (radical sign) of the component that is in the highest degree. Suppose that for the numerator of the original expression this will lead to the appearance of the factor a^p, and for the denominator b^q. Obviously, all the remaining terms have the form С/(n-k) and tend to zero for n>k (n tends to infinity). Then write down the answer: 0 if pq.

Let us indicate a non-traditional way of finding the limit of a sequence and infinite sums. We will use functional sequences (their function members defined on some interval (a,b)). Example 3. Find the sum of the form 1+1/2! +1/3! +…+1/n! +…=s .Solution. Any number a^0=1. Put 1=exp(0) and consider the function sequence (1+x+x^2/2! +x^3/3! +…+x^/n, n=0,1,2,..,n… . Легко заметить, что записанный полином совпадает с многочленом Тейлора по степеням x, который в данном случае совпадает с exp(x). Возьмите х=1. Тогдаexp(1)=e=1+1+1/2! +1/3! +…+1/n! +…=1+s. Ответ s=e-1.!}

Numeric sequence.
How ?

In this lesson, we will learn a lot of interesting things from the life of members of a large community called Vkontakte number sequences. The topic under consideration refers not only to the course of mathematical analysis, but also touches on the basics discrete mathematics. In addition, the material will be required for the development of other sections of the tower, in particular, during the study number series and functional rows. You can tritely say that this is important, you can say reassuringly that it’s simple, you can say a lot more on-duty phrases, but today is the first, unusually lazy school week, so it’s terribly breaking me to write the first paragraph =) I already saved the file in my heart and got ready to sleep, suddenly… the idea of ​​a frank confession lit up the head, which incredibly relieved the soul and pushed for further tapping of the fingers on the keyboard.

Let's digress from summer memories and look into this fascinating and positive world of a new social network:

The concept of a numerical sequence

First, let's think about the word itself: what is a sequence? Consistency is when something is located behind something. For example, the sequence of actions, the sequence of the seasons. Or when someone is located behind someone. For example, a sequence of people in a queue, a sequence of elephants on a path to a watering hole.

Let us immediately clarify the characteristic features of the sequence. First of all, sequence members are located strictly in a certain order. So, if two people in the queue are swapped, then this will already be another subsequence. Secondly, to each sequence member you can assign a serial number:

It's the same with numbers. Let be to each natural value according to some rule mapped to a real number. Then we say that a numerical sequence is given.

Yes, in mathematical problems, in contrast to 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:
is a sequence of positive even numbers:

Thus, the record uniquely determines all members of the sequence - this is the rule (formula) according to which the natural values numbers are matched. Therefore, the sequence is often briefly denoted by a common member, and other Latin letters can be used instead of "x", for example:

Sequence of positive odd numbers:

Another common sequence:

As, probably, many have noticed, the variable "en" plays the role of a kind of counter.

In fact, we dealt with numerical sequences back in middle school. Let's remember arithmetic progression. I will not rewrite the definition, let's touch on the very essence with a specific example. Let be the first term and step arithmetic progression. Then:
is the second term of this progression;
is the third member of this progression;
- fourth;
- fifth;
…
And, obviously, the nth member is asked recurrent formula

Note : in a recursive formula, each next term is expressed in terms of the previous term or even in terms of a whole set of previous terms.

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 is derived: . In our case:

Substitute natural numbers in the formula and check the correctness of the numerical sequence constructed above.

Similar calculations can be made for geometric progression, the nth term of which is given by the formula , where is the first term , and is denominator progressions. In matan assignments, the first term is often equal to one.

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 -1, and to an even power is one.

The progression is called infinitely decreasing, if (last two cases).

Let's add two new friends to our list, one of which has just knocked on the monitor matrix:

The sequence in mathematical jargon is called a "flasher":

Thus, sequence members can be repeated. So, in the considered example, the sequence consists of two infinitely alternating numbers.

Does it happen that the sequence consists of the same numbers? Certainly. For example, it sets an infinite number of "triples". For aesthetes, there is a case when “en” still formally appears in the formula:

Let's invite a simple girlfriend to dance:

What happens when "en" increases to infinity? Obviously, the terms of the sequence will infinitely close approach zero. This is the limit of this sequence, which is written as follows:

If the limit of a sequence is 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 analyze its meaning:

Let us depict the terms of the sequence and the neighborhood symmetric with respect to zero (limit) on the real line:


Now hold the blue neighborhood with the edges of your palms and start to reduce it, pulling it to the limit (red dot). A number is the limit of a sequence if FOR ANY pre-selected -neighborhood (arbitrarily small) inside it will be infinitely many members of the sequence, and OUTSIDE of it - only final number of members (or none at all). That is, the epsilon neighborhood can be microscopic, and even less, but the “infinite tail” of the sequence must sooner or later fully enter this area.

The sequence is also infinitely small: 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, an elementary example:

Here the fraction tends to zero, and accordingly, the limit is equal to "two".

If the sequence there is a finite limit, then it is called converging(in particular, infinitesimal at ). Otherwise - divergent, while 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 move steadily towards "plus infinity":

An arithmetic progression with the first term and a step is also infinitely large:

By the way, any arithmetic progression also diverges, except for the case with a zero step - when infinitely added to a specific number. The limit of such a sequence exists and coincides with the first term.

Sequences have a similar fate:

Any infinitely decreasing geometric progression, as the name implies, infinitely small:

If the denominator is a geometric progression, then the sequence is infinitely largeA:

If, for example, , then there is no limit at all, since the members tirelessly jump either to “plus infinity”, then to “minus infinity”. And common sense and matan's theorems suggest that if something strives somewhere, then this cherished place is unique.

After a little revelation it becomes clear that the flasher is to blame for the unrestrained throwing, which, by the way, diverges on its own.
Indeed, for a sequence it is easy to choose a -neighbourhood, which, say, clamps only the number -1. As a result, an infinite number of sequence members (“plus ones”) will remain outside the given neighborhood. But by definition, the "infinite tail" of the sequence from a certain moment (natural number) must fully enter ANY neighborhood of its limit. Conclusion: there is no limit.

Factorial is infinitely large sequence:

Moreover, it grows by leaps and bounds, so it is a number that has more than 100 digits (digits)! Why exactly 70? It asks for mercy my engineering calculator.

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:

But now it is necessary to be able to solve the limits of functions, at least at the level of two basic lessons: Limits. Solution examples and Remarkable 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"– in the direction of 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 separate isolated members. One, two, three, four, five, the bunny went out for a walk. The argument of the function is characterized by continuity, that is, “x” smoothly, without incident, tends to one or another value. And, accordingly, the values ​​of the function will also continuously approach their limit.

Because of discreteness within the sequences there are their own branded things, such as factorials, flashers, progressions, etc. And now I will try to analyze the limits that are characteristic of sequences.

Let's start with progressions:

Example 1

Find the limit of a sequence

Decision: something similar to an infinitely decreasing geometric progression, but is it really? For clarity, we write out the first few terms:

Since , we are talking about sum members of an infinitely decreasing geometric progression, which is calculated by the formula .

Making a decision:

We use the formula for the sum of an infinitely decreasing geometric progression: . In this case: - the first term, - the denominator of the progression.

Example 2

Write the first four terms of the sequence and find its limit

This is a do-it-yourself example. 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 is the nth term of the progression.

Since 'en' always tends to 'plus infinity' within sequences, it's not surprising that indeterminacy is one of the most popular.
And many examples are solved in exactly the same way as the limits of functions!

Or maybe something more complicated like ? Check out Example #3 of the article Limit Solving Methods.

From a formal point of view, the difference will be only in one letter - there is “x”, and here “en”.
The reception is the same - the numerator and denominator must be divided by "en" in the highest degree.

Also, within sequences, uncertainty is quite common. You can learn how to solve limits like from Examples No. 11-13 of the same article.

To deal with the limit, refer to Example #7 of the lesson Remarkable Limits(the second remarkable limit is also valid for the discrete case). The solution will again be like a carbon copy with a difference in a single letter.

The following four examples (Nos. 3-6) are also “two-faced”, but in practice, for some reason, they are more typical for the limits of sequences than for the limits of functions:

Example 3

Find the limit of a sequence

Decision: first complete solution, then step by step comments:

(1) In the numerator we use the formula twice.

(2) We give like terms in the numerator.

(3) To eliminate uncertainty, we divide the numerator and denominator by ("en" in the highest degree).

As you can see, nothing complicated.

Example 4

Find the limit of a sequence

This is an example for a do-it-yourself solution, abbreviated multiplication formulas to help.

Within s demonstrative sequences use a similar method of dividing the numerator and denominator:

Example 5

Find the limit of a sequence

Decision let's do it the same way:

A similar theorem is also true, by the way, for functions: the product of a bounded function by an infinitesimal function is an infinitesimal function.

Example 9

Find the limit of a sequence

Number Sequence Limit is the limit of the sequence of elements of the number space. A number space is a metric space in which distance is defined as the modulus of the difference between elements. Therefore, the number is called sequence limit, if for any there exists a number depending on such that the inequality holds for any .

The concept of the limit of a sequence of real numbers is formulated quite simply, and in the case of complex numbers, the existence of a limit of a sequence is equivalent to the existence of limits of the corresponding sequences of real and imaginary parts of complex numbers.

The limit (of a numerical sequence) is one of the basic concepts of mathematical analysis. Each real number can be represented as the limit of a sequence of approximations to the desired value. The number system provides such a sequence of refinements. Integer irrational numbers are described by periodic sequences of approximations, while irrational numbers are described by non-periodic sequences of approximations.

In numerical methods, where the representation of numbers with a finite number of signs is used, the choice of the system of approximations plays a special role. The criterion for the quality of the system of approximations is the rate of convergence. In this respect, representations of numbers in the form of continued fractions are effective.

Definition

The number is called the limit of the numerical sequence, if the sequence is infinitely small, i.e., all its elements, starting from some, are less than any positive number taken in advance.

In the event that a numerical sequence has a limit in the form of a real number, it is called converging to this number. Otherwise, the sequence is called divergent . If, moreover, it is unlimited, then its limit is assumed to be equal to infinity.

In addition, if all elements of an unbounded sequence, starting from some number, have a positive sign, then we say that the limit of such a sequence is equal to plus infinity .

If the elements of an unlimited sequence, starting from some number, have a negative sign, then they say that the limit of such a sequence is equal to minus infinity .

This definition has an unavoidable shortcoming: it explains what a limit is, but does not give a way to calculate it, nor information about its existence. All this is deduced from the properties of the limit proved below.