Sequence limit solution examples. Sequence limit - basic theorems and properties

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. Solution. 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.

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.!}

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 some 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. Pozniak. Fundamentals of mathematical analysis. Part 1. Moscow, 2005.

See also:

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.

Today at the lesson we will analyze strict sequencing And strict definition of the limit of a function, as well as learn how to solve the corresponding problems of a theoretical nature. The article is intended primarily for first-year students of natural sciences and engineering specialties who have begun to study the theory of mathematical analysis and have encountered difficulties in understanding this section of higher mathematics. In addition, the material is quite accessible to high school students.

Over the years of the site’s existence, I received an unkind dozen of letters with approximately the following content: “I don’t understand mathematical analysis well, what should I do?”, “I don’t understand matan at all, I’m thinking of quitting my studies,” etc. Indeed, it is the matan who often thins out the student group after the very first session. Why are things like this? Because the subject is unthinkably complex? Not at all! The theory of mathematical analysis is not so difficult as it is peculiar. And you need to accept and love her for who she is =)

Let's start with the most difficult case. First and foremost, don't drop out of school. Understand correctly, quit, it will always have time ;-) Of course, if in a year or two from the chosen specialty it will make you sick, then yes - you should think about it (and not smack the fever!) about changing activities. But for now it's worth continuing. And, please, forget the phrase “I don’t understand anything” - it doesn’t happen that you don’t understand anything at all.

What to do if the theory is bad? By the way, this applies not only to mathematical analysis. If the theory is bad, then first you need to SERIOUSLY put on practice. At the same time, two strategic tasks are solved at once:

– Firstly, a significant proportion of theoretical knowledge has come about through practice. And so many people understand theory through ... - that's right! No, no, you didn't think about that.

- And, secondly, practical skills are very likely to “stretch” you in the exam, even if ..., but let's not tune in like that! Everything is real and everything is really “lifted” in a fairly short time. Mathematical analysis is my favorite section of higher mathematics, and therefore I simply could not help but lend you a helping hand:

At the beginning of the 1st semester, sequence limits and function limits usually pass. Do not understand what it is and do not know how to solve them? Start with an article Function Limits, in which the concept itself is considered “on the fingers” and the simplest examples are analyzed. Then work through other lessons on the topic, including a lesson about within sequences, on which I have actually already formulated a rigorous definition.

What icons besides inequality signs and modulus do you know?

- a long vertical stick reads like this: “such that”, “such that”, “such that” or “such that”, in our case, obviously, we are talking about a number - therefore “such that”;

- for all "en" greater than ;

– module sign means distance, i.e. this entry tells us that the distance between values is less than epsilon.

Well, is it deadly difficult? =)

After mastering the practice, I am waiting for you in the following paragraph:

Indeed, let's think a little - how to formulate a rigorous definition of a sequence? ... The first thing that comes to mind in the light practical session: "the limit of a sequence is the number to which the members of the sequence approach infinitely close."

Okay, let's write subsequence :

It is easy to grasp that subsequence approach infinitely close to -1, and even-numbered terms - to "unit".

Maybe two limits? But then why can't some sequence have ten or twenty of them? That way you can go far. In this regard, it is logical to assume that if the sequence has a limit, then it is unique.

Note : the sequence has no limit, but two subsequences can be distinguished from it (see above), each of which has its own limit.

Thus, the above definition turns out to be untenable. Yes, it works for cases like (which I did not quite correctly use in simplified explanations of practical examples), but now we need to find a strict definition.

Attempt two: “the limit of a sequence is the number that ALL members of the sequence approach, with the exception of, perhaps, their final quantities." This is closer to the truth, but still not entirely accurate. So, for example, the sequence half of the terms do not approach zero at all - they are simply equal to it =) By the way, the "flashing light" generally takes two fixed values.

The formulation is not difficult to clarify, but then another question arises: how to write the definition in mathematical terms? The scientific world struggled with this problem for a long time until the situation was resolved. famous maestro, which, in essence, formalized the classical mathematical analysis in all its rigor. Cauchy offered to operate surroundings which greatly advanced the theory.

Consider some point and its arbitrary-neighborhood:

The value of "epsilon" is always positive, and moreover, we have the right to choose it ourselves. Assume that the given neighborhood contains a set of terms (not necessarily all) some sequence. How to write down the fact that, for example, the tenth term fell into the neighborhood? Let it be on the right side of it. Then the distance between the points and should be less than "epsilon": . However, if the "x tenth" is located to the left of the point "a", then the difference will be negative, and therefore the sign must be added to it module: .

Definition: a number is called the limit of a sequence if for any its surroundings (preselected) there is a natural number - SUCH that ALL members of the sequence with higher numbers will be inside the neighborhood:

Or shorter: if

In other words, no matter how small the value of "epsilon" we take, sooner or later the "infinite tail" of the sequence will FULLY be in this neighborhood.

So, for example, the "infinite tail" of the sequence FULLY goes into any arbitrarily small -neighborhood of the point . Thus, this value is the limit of the sequence by definition. I remind you that a sequence whose limit is zero is called infinitesimal.

It should be noted that for the sequence it is no longer possible to say "infinite tail will come”- members with odd numbers are in fact equal to zero and “do not go anywhere” =) That is why the verb “will end up” is used in the definition. And, of course, the members of such a sequence as also "do not go anywhere." By the way, check if the number will be its limit.

Let us now show that the sequence has no limit. Consider, for example, a neighborhood of the point . It is quite clear that there is no such number, after which ALL members will be in this neighborhood - odd members will always "jump" to "minus one". For a similar reason, there is no limit at the point .

Fix the material with practice:

Example 1

Prove that the limit of the sequence is zero. Indicate the number , after which all members of the sequence are guaranteed to be inside any arbitrarily small -neighborhood of the point .

Note : for many sequences, the desired natural number depends on the value - hence the notation .

Solution: consider arbitrary will there be number - such that ALL members with higher numbers will be inside this neighborhood:

To show the existence of the required number , we express in terms of .

Since for any value "en", then the modulus sign can be removed:

We use "school" actions with inequalities that I repeated in the lessons Linear inequalities And Function scope. In this case, an important circumstance is that "epsilon" and "en" are positive:

Since on the left we are talking about natural numbers, and the right side is generally fractional, it needs to be rounded:

Note : sometimes a unit is added to the right for reinsurance, but in fact this is an overkill. Relatively speaking, if we also weaken the result by rounding down, then the nearest suitable number (“three”) will still satisfy the original inequality.

And now we look at inequality and remember that initially we considered arbitrary-neighborhood, i.e. "epsilon" can be equal to anyone positive number.

Output: for any arbitrarily small -neighborhood of the point, the value . Thus, a number is the limit of a sequence by definition. Q.E.D.

By the way, from the result a natural pattern is clearly visible: the smaller the -neighborhood, the greater the number after which ALL members of the sequence will be in this neighborhood. But no matter how small the "epsilon" is, there will always be an "infinite tail" inside, and outside - even if it is large, however final number of members.

How are the impressions? =) I agree that it is strange. But strictly! Please re-read and think again.

Consider a similar example and get acquainted with other techniques:

Example 2

Solution: by the definition of a sequence, it is necessary to prove that (Speak out loud!!!).

Consider arbitrary-neighborhood of the point and check, does it exist natural number - such that for all larger numbers the following inequality holds:

To show the existence of such an , you need to express "en" through "epsilon". We simplify the expression under the module sign:

The module destroys the minus sign:

The denominator is positive for any "en", therefore, the sticks can be removed:

Shuffling:

Now we should take the square root, but the catch is that for some "epsilons" the right side will be negative. To avoid this trouble let's strengthen inequality modulus:

Why can this be done? If, relatively speaking, it turns out that , then the condition will be satisfied even more so. The module can just increase wanted number , and that will suit us too! Roughly speaking, if the hundredth is suitable, then the two hundredth will do! According to the definition, you need to show the very existence of the number(at least some), after which all members of the sequence will be in -neighbourhood. By the way, that is why we are not afraid of the final rounding of the right side up.

Extracting the root:

And round the result:

Output: because the value of "epsilon" was chosen arbitrarily, then for any arbitrarily small -neighborhood of the point, the value , such that the inequality . In this way, by definition. Q.E.D.

I advise especially understand the strengthening and weakening of inequalities - these are typical and very common methods of mathematical analysis. The only thing you need to monitor the correctness of this or that action. So, for example, the inequality by no means loosen, subtracting, say, one:

Again, conditional: if the number fits exactly, then the previous one may no longer fit.

The following example is for a standalone solution:

Example 3

Using the definition of a sequence, prove that

Short solution and answer at the end of the lesson.

If the sequence infinitely great, then the definition of the limit is formulated in a similar way: a point is called the limit of a sequence if for any, arbitrarily large there is a number such that for all larger numbers , the inequality will be satisfied. The number is called the neighborhood of the point "plus infinity":

In other words, no matter how large the value we take, the “infinite tail” of the sequence will necessarily go into the -neighborhood of the point , leaving only a finite number of terms on the left.

Working example:

And an abbreviated notation: if

For the case, write the definition yourself. The correct version is at the end of the lesson.

After you have "filled" your hand with practical examples and figured out the definition of the limit of a sequence, you can turn to the literature on mathematical analysis and / or your lecture book. I recommend downloading the 1st volume of Bohan (easier - for part-time students) and Fikhtengoltz (more detailed and thorough). Of the other authors, I advise Piskunov, whose course is focused on technical universities.

Try to conscientiously study the theorems that concern the limit of the sequence, their proofs, consequences. At first, the theory may seem "cloudy", but this is normal - it just takes some getting used to. And many will even get a taste!

Strict definition of the limit of a function

Let's start with the same thing - how to formulate this concept? The verbal definition of the limit of a function is formulated much more simply: “a number is the limit of a function, if with “x” tending to (both left and right), the corresponding values of the function tend to » (see drawing). Everything seems to be normal, but words are words, meaning is meaning, an icon is an icon, and strict mathematical notation is not enough. And in the second paragraph, we will get acquainted with two approaches to solving this issue.

Let the function be defined on some interval except, possibly, for the point . In the educational literature, it is generally accepted that the function there not defined:

This choice highlights the essence of the function limit: "x" infinitely close approaches , and the corresponding values of the function are infinitely close to . In other words, the concept of a limit does not imply an “exact approach” to points, namely infinitely close approximation, it does not matter whether the function is defined at the point or not.

The first definition of the limit of a function, not surprisingly, is formulated using two sequences. Firstly, the concepts are related, and secondly, the limits of functions are usually studied after the limits of sequences.

Consider the sequence points (not on the drawing) belonging to the interval and other than, which converges to . Then the corresponding values of the function also form a numerical sequence, the members of which are located on the y-axis.

Heine function limit for any point sequences (belonging to and different from), which converges to the point , the corresponding sequence of function values converges to .

Eduard Heine is a German mathematician. ... And there is no need to think anything like that, there is only one gay in Europe - this is Gay-Lussac =)

The second definition of the limit was built ... yes, yes, you are right. But first, let's look at its design. Consider an arbitrary -neighbourhood of the point ("black" neighborhood). Based on the previous paragraph, the notation means that some value function is located inside the "epsilon"-environment.

Now let's find -neighborhood that corresponds to the given -neighborhood (mentally draw black dotted lines from left to right and then from top to bottom). Note that the value is chosen along the length of the smaller segment, in this case, along the length of the shorter left segment. Moreover, the "crimson" -neighborhood of a point can even be reduced, since in the following definition the very fact of existence is important this neighbourhood. And, similarly, the entry means that some value is inside the "delta" neighborhood.

Cauchy limit of a function: the number is called the limit of the function at the point if for any preselected neighborhood (arbitrarily small), exists-neighborhood of the point , SUCH that: AS ONLY values (owned) included in this area: (red arrows)- SO IMMEDIATELY the corresponding values of the function are guaranteed to enter the -neighborhood: (blue arrows).

I must warn you that in order to be more intelligible, I improvised a little, so do not abuse it =)

Shorthand: if

What is the essence of the definition? Figuratively speaking, by infinitely decreasing the -neighbourhood, we "accompany" the values of the function to its limit, leaving them no alternative to approach somewhere else. Quite unusual, but again strictly! To get the idea right, reread the wording again.

! Attention: if you need to formulate only definition according to Heine or only Cauchy definition please don't forget about significant preliminary comment: "Consider a function that is defined on some interval except perhaps a point". I stated this once at the very beginning and did not repeat it each time.

According to the corresponding theorem of mathematical analysis, the Heine and Cauchy definitions are equivalent, but the second variant is the most well-known (still would!), which is also called the "limit on the tongue":

Example 4

Using the definition of a limit, prove that

Solution: the function is defined on the entire number line except for the point . Using the definition of , we prove the existence of a limit at a given point.

Note : the magnitude of the "delta" neighborhood depends on the "epsilon", hence the designation

Consider arbitrary-neighborhood. The task is to use this value to check whether does it exist- neighborhood, SUCH, which from the inequality follows the inequality .

Assuming that , we transform the last inequality:
(decompose the square trinomial)

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:

All limits are abbreviated as lim.
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:

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.
If a series of numbers has a limit, then the sequence of these numbers is limited.
The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
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

Limits of sequences are the same significant (in most cases) value 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:

The sum of any number of arbitrarily small quantities will also be a small quantity.
The sum of any number of large values will be an infinitely large value.
The product of arbitrarily small quantities is infinitely small.
The product of arbitrarily large numbers is an infinitely large quantity.
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.