Introduction

numerical Cauchy d'Alembert

The concept of infinite sums was actually known to scientists Ancient Greece(Eudoxus, Euclid, Archimedes). Finding infinite sums was an integral part of the so-called exhaustion method, widely used by ancient Greek scientists to find the areas of figures, volumes of bodies, lengths of curves, etc. So, for example, Archimedes, to calculate the area of ​​a parabolic segment (i.e., a figure bounded by a straight line and a parabola), found the sum of an infinite geometric progression with a denominator of 1/4.

Mathematicians began to use series as an independent concept in the 17th century. I. Newton and G. Leibniz used series to solve algebraic and differential equations. The theory of series in the 18th-19th centuries. developed in the works of J. and I. Bernoulli, B. Taylor, C. Maclaurin, L. Euler, J. d'Alembert, J. Lagrange and others. A rigorous theory of series was created in the 19th century. based on the concept of limit in the works of K. Gauss, B. Bolzano, O. Cauchy, P. Dirichlet, N. Abel, K. Weierstrass, B. Riemann and others.

The relevance of studying this problem is due to the fact that the branch of mathematics that makes it possible to solve any well-posed problem with sufficient accuracy for practical use is called series theory. Even if some subtle concepts mathematical analysis appeared out of connection with the theory of series, they were immediately applied to series, which served as a kind of tool for testing the significance of these concepts. This situation continues to this day. Thus, it seems relevant to study number series, their basic concepts and features of series convergence.

1. History

.1 First mention and use of the number series

The rules of arithmetic give us the ability to determine the sum of two, three, four and, in general, any finite set of numbers. What if the number of terms is infinite? Even if it is the “smallest” infinity, i.e. let the number of terms be countable.

Finding infinite sums was an integral part of the so-called exhaustion method, widely used by ancient Greek scientists to find the areas of figures, volumes of bodies, lengths of curves, etc. So, for example, Archimedes, to calculate the area of ​​a parabolic segment (i.e., a figure bounded by a straight line and a parabola), found the sum of an infinite geometric progression with a denominator of 1/4.

Almost two and a half thousand years ago, the Greek mathematician and astronomer Eudoxus of Cnidus used the method of “exhaustion” to find areas and volumes. The idea of ​​this method is to divide the body under study into a countable number of parts, the areas or volumes of which are known, and then add these volumes. This method was used by both Euclid and Archimedes. Naturally, there was no complete and accurate justification of the method in the works of ancient mathematicians. Before this, it was necessary to go through a long two-thousand-year journey, on which there were brilliant revelations, mistakes, and curiosities.

Here, for example, is how one medieval theologian reasoned when proving - no more and no less - the existence Almighty God.

Let us write S in equal quantities as an infinite sum

S = 1010101010… (1)

“Let us replace each zero on the right side of this equality with the sum 1+(-1)

S =1+(-1)+ 1+(-1)+ 1+(-1)+… (2)

Leaving the first term on the right side of (2) alone, we use brackets to combine the second term with the third, the fourth with the fifth, etc. Then

S=1 + ((-1) +1) + ((-1) +1) +… = 1+0+0+… = 1.”

“If you can get one from zero at will, then the assumption of creating the world from nothing is also acceptable!”

Do we agree with this reasoning? Of course not. From the point of view of modern mathematics, the author’s mistake is that he tries to operate with concepts that are not given a definition (what is it - “the sum of an infinite number of terms”), and makes transformations (opening brackets, regrouping), the legality of which is not was justified by him.

The greatest mathematicians of the 17th and 18th centuries - Isaac Newton (1642-1727), Gottfried Wilhelm Leibniz (1646-1716), Brooke Taylor (1685-1731) - widely used counting sums without paying enough attention to the question of what exactly this concept means. ), Colin Maclaurin (1698-1746), Joseph Louis Lagrange (1736-1813). Leonard and Euler (1707-1783) were noted for their virtuosic mastery of handling rows, but at the same time he often admitted the insufficient justification for the techniques he used. A hundred papers repeatedly contain sentences like this: “We found that these two infinite expressions are equal, although it turned out to be impossible to prove it.” He warns mathematicians against using “divergent series,” although he himself did not always care about this, and only brilliant intuition protects him from incorrect conclusions; True, he also has “punctures”.

By the beginning of the 19th century, the need for a careful justification of the properties of “counting sums” became clear. In 1812, Carl Friedrich Gauss (1777-1865) gave the first example of the study of series convergence; in 1821, our good friend Augustin Louis Cauchy (1789-1857) established the basic modern principles of series theory.

.2 Further study of number series. A clear formulation of the concept of a number series

The summation of infinite geometric progressions with a denominator less than 1 was already carried out in ancient times (Archimedes). The divergence of the harmonic series was established by the Italian scientist Mengoli in 1650. Power series appeared in Newton (1665), who believed that any function could be represented by a power series. Scientists of the 18th century constantly encountered series in calculations, but attention was not always paid to the issue of convergence. The exact theory of series begins with the works of Gauss (1812), Bolzano (1817) and, finally, Cauchy, where the modern definition of the sum of a convergent series was first given and the main theorems were established. In 1821, Cauchy published the “Course of Analysis at the Royal Polytechnic School”, which had highest value to disseminate new ideas for substantiating mathematical analysis in the first half of the 19th century.

"Next is an unlimited sequence of quantities

resulting from one another according to a certain law... Let

is the sum of the first n terms, where n is any integer. If, with a constant increase in the values ​​of n, the sum indefinitely approaches a known limit S, the series is called convergent, and this limit is the sum of the series. On the contrary, if with an unlimited increase in n the sum does not approach any a certain limit, the series will be divergent and will not have a sum..." [From the first part of the "Course of Analysis at the Royal Polytechnic School" by O. Cauchy (1821) ( No. 54 vol. III, p. 114-116, translation by A.P. Yushkevich}]

.3 Problems leading to the concept of a number series and those in which it was used

Swift-footed Achilles will never catch up with the tortoise if at the beginning of the movement the tortoise was some distance ahead of him. Indeed, let the initial distance be a and let Achilles run k times faster than the tortoise. When Achilles passes distance a, the tortoise crawls away to a/k, when Achilles passes this distance, the tortoise crawls away to a/, etc., i.e. each time there will be a non-zero distance between the competitors.

In this aporia, in addition to the same difficulty of counted infinity, there is one more thing. Suppose that at some point in time Achilles catches up with the tortoise. Let's write down the path of Achilles

and the way of the turtle

Each segment of the path a/ traversed by Achilles corresponds to a segment of the path a/ of the tortoise. Therefore, by the time of the meeting, Achilles must have covered “as many” sections of the path as the tortoise. On the other hand, each segment a/ traversed by the tortoise can be associated with an equal segment of Achilles’ path. But, in addition, Achilles must run one more segment of length a, i.e. he must travel one more segment than the turtle. If the number of segments covered by the last one is b, then we get

"Arrow". "Arrow". If time and space consist of indivisible particles, then a flying arrow is motionless, since at each indivisible moment of time it occupies an equal position, i.e. is at rest, and a period of time is the sum of such indivisible moments.

This aporia is directed against the idea of continuous value- as the sum of an infinite number of indivisible particles.

"Stadium". Let equal masses move across the stadium along parallel straight lines at equal speeds, but in opposite directions. Let row mean stationary masses, row mean masses moving to the right, and row mean masses moving to the left (Fig. 1). Let us now consider the masses. as indivisible. At an indivisible moment of time, an indivisible part of space passes through. Indeed, if at an indivisible moment of time a certain body passed through more than one indivisible part of space, then the indivisible moment of time would be divisible, but if less, then the indivisible part of space could be divided. Let us now consider the movement of indivisibles relative to each other: in two indivisible moments of time, two indivisible parts will pass, and at the same time count four indivisible parts, i.e. an indivisible moment of time will turn out to be divisible.

This aporia can be given a slightly different form. In the same time t, a point passes half a segment and a whole segment. But each indivisible moment of time corresponds to an indivisible part of space traversed during this time. Then a certain segment a and segment 2a contain the “same” number of points, “same” in the sense that a one-to-one correspondence can be established between the points of both segments. This was the first time such a correspondence was established between points of segments of different lengths. If we assume that the measure of a segment is obtained as the sum of indivisible measures, then the conclusion is paradoxical.

2. Application of the number series

.1 Definition

Let an infinite number sequence be given

Definition 1.1. Number series or just near is called an expression (sum) of the form

The numbers are called members of a number, - general or nth member of the series.

To define the series (1.1), it is enough to specify the function of the natural argument of calculating the th term of the series by its number

From the terms of series (1.1) we form a numerical sequence of partials amounts where is the sum of the first terms of the series, which is called n-th partial amount, i.e.

…………………………….

…………………………….

A numerical sequence with an unlimited increase in number can:

) have a finite limit;

) have no finite limit (the limit does not exist or is equal to infinity).

Definition 1.2. Series (1.1) is called convergent, if the sequence of its partial sums (1.5) has a finite limit, i.e.

In this case the number is called amount series (1.1) and is denoted

Definition 1.3. Series (1.1) is called divergent, if the sequence of its partial sums does not have a finite limit.

No sum is assigned to the divergent series.

Thus, the problem of finding the sum of a convergent series (1.1) is equivalent to calculating the limit of the sequence of its partial sums.

.2 Basic properties of number series

The properties of a sum of a finite number of terms differ from the properties of a series, i.e. the sum of an infinite number of terms. So, in the case of a finite number of terms, they can be grouped in any order, this will not change the sum. There are convergent series (conditionally convergent), for which, as Riemann Georg Friedrich Bernhard showed, by properly changing the order of their terms, you can make the sum of the series equal to any number, and even a divergent series.

Example 2.1.Consider a divergent series of the form

By grouping its members in pairs, we obtain a convergent number series with a sum equal to zero:

On the other hand, by grouping its terms in pairs, starting with the second term, we also obtain a convergent series, but with a sum equal to one:

Convergent series have certain properties that make it possible to treat them as if they were finite sums. So they can be multiplied by numbers, added and subtracted term by term. They can combine any adjacent terms into groups.

Theorem 2.1.(A necessary sign of convergence of a series).

If series (1.1) converges, then its common term tends to zero as n increases indefinitely, i.e.

The proof of the theorem follows from the fact that, and if

S is the sum of series (1.1), then

Condition (2.1) is a necessary but not sufficient condition for the convergence of the series. That is, if the common term of a series tends to zero at, this does not mean that the series converges. For example, for the harmonic series (1.2), however, it diverges.

Consequence(A sufficient sign of the divergence of the series).

If the common term of a series does not tend to zero at, then this series diverges.

Property 2.1. The convergence or divergence of a series will not change if a finite number of terms are arbitrarily removed from it, added to it, or rearranged in it (in this case, for a convergent series, its sum may change).

The proof of the property follows from the fact that series (1.1) and any of its remainders converge or diverge simultaneously.

Property 2.2. A convergent series can be multiplied by a number, i.e., if the series (1.1) converges, has the sum S and c is a certain number, then

The proof follows from the fact that the following equalities hold for finite sums:

Property 2.3. Convergent series can be added and subtracted term by term, i.e. if the rows

converge,

converges and its sum is equal to i.e.

The proof follows from the properties of the limit of finite sums, i.e.

Comparison sign

Let two positive series be given

and the conditions are met for all n=1,2,…

Then: 1) from the convergence of series (3.2) follows the convergence of series (3.1);

) from the divergence of series (3.1) follows the divergence of series (3.2).

Proof. 1. Let the series (3.2) converge and its sum equal to B. The sequence of partial sums of the series (3.1) is non-decreasing bounded above by the number B, i.e.

Then, due to the properties of such sequences, it follows that it has a finite limit, i.e. series (3.1) converges.

Let series (3.1) diverge. Then, if series (3.2) converges, then by virtue of point 1 proved above, the original series would also converge, which contradicts our condition. Consequently, series (3.2) also diverges.

This criterion is convenient to apply to determining the convergence of series, comparing them with series whose convergence is already known.

D'Alembert's sign

Then: 1) at q< 1 ряд (1.1) сходится;

) for q > 1, series (1.1) diverges;

) for q = 1 nothing can be said about the convergence of series (1.1); additional research is needed.

Comment: Series (1.1) will also diverge in the case when

Cauchy's sign

Let the terms of the positive series (1.1) be such that there is a limit

Then: 1) at q< 1 ряд (1.1) сходится;

) for q > 1, series (1.1) diverges;

3) for q = 1, nothing can be said about the convergence of series (1.1); additional research is needed.

Integral Cauchy-Maclaurin test

Let the function f(x) be a continuous non-negative non-increasing function on the interval

Then the series and the improper integral converge or diverge simultaneously.

.3 Objectives

Number series are used not only in mathematics, but also in a number of other sciences. I would like to give a few examples of such use.

For example, to study the properties of clastic rock structures. In practice, the use of the concept of “structure” has mainly been reduced to characterizing the dimensional parameters of grains. In this regard, the concept of “structure” in petrography does not correspond to the concept of “structure” in crystallography, structural geology and other sciences about the structure of matter. In the latter, “structure” is more consistent with the concept of “texture” in petrography and reflects the way space is filled. If we accept that “structure” is a spatial concept, then the following structures should be considered meaningless: secondary or primary structures and textures; crystalline, chemical, substitutions (corrosion, recrystallization, etc.), deformation structures, oriented, residual structures, etc. Therefore, these “structures” are called “false structures.”

Structure is a set of structural elements characterized by grain sizes and their quantitative relationships.

When carrying out specific classifications, linear grain parameters with the sequence

Although quantitative estimates prevalence is carried out through area (percentage) parameters. This sequence can be of considerable length and is never built. Usually they only talk about the limits of parameter variation, naming the maximum (max) and minimum (min) values ​​of grain sizes.

One of the directions for representing P4 is the use of number series, which are constructed in the same way as the above sequence, but instead of (?) a sum sign (+) is placed. The convolution of all sequences is carried out by combining equal elements and adding their areas. Then we have the sequence:

The expression means that the area occupied by all sections of those grains i whose size is equal is measured.

This feature of the grains allows for a numerical analysis of the obtained relationships. Firstly, the parameter can be considered as the values ​​of the coordinate axis and thus build some graph S=f(l). Secondly, the sequence (RSl) 1 can be ranked, for example, in descending order of coefficients, resulting in a series

It is this series that is called the structure of a given rock section, and it is also the definition of the concept “structure”. The parameter is an element of the structure, and the parameter k= is the length of the structure. By construction n=k. This representation of the structure allows comparison of different structures with each other.

Also, Kirill Pavlovich Butusov discovered the phenomenon of “resonance of beat waves”, on the basis of which he formulated the “law of planetary periods”, due to which the periods of revolutions of the planets form the Fibonacci and Lucas number series and proved that the “law of planetary distances” of Johann Titius is a consequence of “ resonance of beat waves" (1977). At the same time, he discovered the manifestation of the “golden section” in the distribution of a number of other parameters of bodies in the Solar System (1977). In this regard, he is working to create “golden mathematics” - a new number system based on Phidias’ number (1.6180339), more adequate to the problems of astronomy, biology, architecture, aesthetics, music theory, etc.

From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, with the help of this Fibonacci series, found a pattern and order in the distances between planets solar system.

However, one case that seemed to contradict the law: there was no planet between Mars and Jupiter. Focused observation of this part of the sky led to the discovery of the asteroid belt. This happened after the death of Titius at the beginning of the 19th century. The Fibonacci series is widely used: it is used to represent the architectonics of living beings, man-made structures, and the structure of Galaxies. These facts are evidence of the independence of the number series from the conditions of its manifestation, which is one of the signs of its universality.

Cryptography is the science of mathematical methods ensuring confidentiality (impossibility of reading information by outsiders) and authenticity (integrity and authenticity of authorship, as well as the impossibility of refusing authorship) of information. The vast majority of modern cryptographic systems use either stream or block algorithms based on various types substitution and permutation ciphers. Unfortunately, almost all algorithms used in stream cryptosystems are intended for use in military and government communications systems, as well as, in some cases, for protecting commercial information, which quite naturally makes them secret and inaccessible for review. The only standard stream encryption algorithms are the American DES standard (CFB and OFB modes) and the Russian GOST 28147-89 standard (gamming mode). However, the stream encryption algorithms used in these standards are classified.

The basis for the functioning of stream cryptosystems are generators of random or pseudo-random sequences. Let's consider this issue in more detail.

Pseudo-random sequences

Secret keys are the basis of cryptographic transformations, for which, following Kerckhoff's rule, the strength of a good encryption system is determined only by the secrecy of the key. However, in practice, creating, distributing and storing keys has rarely been a technically complex, albeit expensive, task. The main problem of classical cryptography has long been the difficulty of generating unpredictable long binary sequences using a short random key. To solve this problem, generators of binary pseudorandom sequences are widely used. Significant progress in the development and analysis of these generators was achieved only in the early sixties. Therefore, this chapter discusses the rules for obtaining keys and generating, based on them, long pseudo-random sequences used by cryptographic systems to convert a message into encryption.

Random or pseudo-random series of numbers obtained programmatically from a key are called gamma in the jargon of domestic cryptographers, after the name y - the letter of the Greek alphabet, which denotes random variables in mathematical notations. It is interesting to note that in the book “Strangers on a Bridge,” written by intelligence officer Abel’s lawyer, the term gamma is given, which CIA specialists commented on “musical exercise?”, That is, in the fifties they did not know its meaning. Obtaining and reproducing implementations of real random series is dangerous, difficult and expensive. Physical modeling of randomness using physical phenomena such as radioactive radiation, shot noise in a vacuum tube, or tunneling breakdown of a semiconductor zener diode does not produce true random processes. Although there are known cases of their successful use in key generation, for example, in the Russian cryptographic device KRYPTON. Therefore, instead of physical processes, computer programs are used to generate gamma, which, although called random number generators, actually produce deterministic number series that only seem random in their properties. They are required to ensure that, even knowing the law of formation, but not knowing the key in the form of initial conditions, no one would be able to distinguish a number series from a random one, as if it were obtained by throwing ideal dice. We can formulate three main requirements for a cryptographically secure generator of a pseudo-random sequence or gamma:

The gamma period must be large enough to encrypt messages of varying lengths.

Gamma should be difficult to predict. This means that if the type of generator and a piece of the gamma are known, then it is impossible to predict the next bit of the gamma after this piece with a probability higher than x. If a cryptanalyst knows any part of the gamut, he will still not be able to determine the bits preceding or following it.

Generating a range should not be associated with major technical and organizational difficulties.

Fibonacci sequences

An interesting class of random number generators has been repeatedly proposed by many experts in integer arithmetic, in particular George Marsalia and Arif Zeiman. Generators of this type are based on the use of Fibonacci sequences. A classic example of such a sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34...). With the exception of its first two terms, each subsequent term is equal to the sum of the two previous ones. If you take only the last digit of each number in the sequence, you will get a sequence of numbers (0, 1, 1, 2, 5, 8, 3, 1, 4, 5, 9, 4...) If this sequence is used to initially fill a large array , then, using this array, you can create a Fibonacci random number generator with a delay, where not neighboring, but distant numbers are added. Marsalia and Zeiman proposed introducing a “carry bit” into the Fibonacci circuit, which can have an initial value of 0 or 1. The “carry addition” generator built on this basis acquires interesting properties; based on them, it is possible to create sequences whose period is significantly greater than that of currently used congruent generators. According to Marsalia's figurative expression, generators of this class can be considered as amplifiers of randomness. “You take a random seed that is several thousand bits long and generate long sequences of random numbers.” However, a long period in itself is not a sufficient condition. Weaknesses in scales can be difficult to detect and require the analyst to use sophisticated sequence analysis techniques to highlight certain patterns that are hidden in a large array of numbers.

Conclusions

Series are widely used in mathematics and its applications, in theoretical research, and in approximate numerical solutions of problems. Many numbers can be written in the form of special series, with the help of which it is convenient to calculate their approximate values ​​with the required accuracy. The series expansion method is effective method studying functions. It is used to calculate approximate values ​​of functions, to calculate and evaluate integrals, to solve all kinds of equations (algebraic, differential, integral).

References

1. Shilov G.E. Mathematical analysis. Functions of one variable. Part 1-2 - M.: Nauka, 1969

Maikov E.V. Mathematical analysis. Number series/E.V. Maikov. - 1999

.“Course of Analysis at the Royal Polytechnic School”

O. Cauchy (1821) (No. 54 vol. III, p. 114-116, translation by A.P. Yushkevich)

History of mathematics from ancient times to early XIX century (edited by A.P. Yushkevich, volume I)

Reader on the history of mathematics (part II) (edited by A.P. Yushkevich)

Higher mathematics: General course: Textbook. - 2nd ed., / A.I. Yablonsky, A.V. Kuznetsov, E.I. Shilkina and others; Under general ed. S.A. Samal. - Mn.: Higher. school, 2000. - 351 p.

Markov L.N., Razmyslovich G.P. Higher mathematics. Part 2. Fundamentals of mathematical analysis and elements of differential equations. - Mn.: Amalthea, 2003. - 352 p.

8. Makarov V.P. Questions of theoretical geology. 7. Elements of the theory of structures. / Contemporary issues and ways to solve them in science, transport, production and education 2007. Odessa, Chernomorye, 2007. T.19. pp. 27 - 40.

9. Polovinkina Yu. Ir. Structures rocks. Part 1: Igneous Rocks; Part 2: Sedimentary Rocks; Part 3: Metamorphic rocks. - M.: Gosgeolizdat, 1948.

10.http://shaping.ru/mku/butusov.asp

Http://www.abc-people.com/idea/zolotsech/gr-txt.htm

Educational and methodological complex of the discipline “Mathematics”. Section 10 "Rows". Theoretical foundations. Guidelines for students. Materials for independent work students. - Ufa: Publishing House USNTU, 2007. - 113 p.

13.http://cryptolog.ru/? Psevdosluchainye_posledovatelmznosti

14. Galuev G.A. Mathematical foundations of cryptology: Educational and methodological manual. Taganrog: TRTU Publishing House 2003.-120 p.

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Consider an infinite sequence of numbers, i.e. a set of numbers in which each natural number n according to a certain rule, a certain number corresponds a n. An expression of the form is called a number series, the numbers themselves are called members of the series, - common member of the series. Briefly the series is written as follows: .

Amounts that contain only n the first members of the series are called partial sums of a series.

A number series is said to be convergent if the sequence of its partial sums has a finite limit. Number S is called the sum of the series.

If the limit does not exist, then the series is said to be divergent.

Example 1. An infinite geometric progression is given. Let's make a series

and examine it for convergence, based on the definition of convergence of a series. To do this, let's make a partial sum =. From the school mathematics course it is known that. Let's remember how this works. To prove this, let's divide

Let us now calculate the limit, taking into account that three cases are possible here:

2) if q= 1, then = and ,

3) if q= -1, then =, and , a = , and . This means that the sequence of partial sums does not have a single limit.

Therefore, we conclude: a geometric progression converges if and diverges at .

Example 2. Prove the divergence of the series

Solution. Let's estimate the partial sum of the series:

> , i.e. > ,

and the limit of the partial sum is equal to infinity (according to the well-known theorem about limits: if x n > y n, then ): = ¥. This means that this series diverges.

Properties of convergent series

Consider two rows and . The second row is obtained from the first by discarding the first m its members. This series is called the remainder of the series and is denoted r n.

Theorem 1. If the terms of a convergent series are multiplied by a certain number WITH, then the convergence of the series will not be violated, and the sum will be multiplied by WITH.

Theorem 2. Two convergent series can be added (subtracted) term by term and the sum of the resulting series will be equal to , where is the sum of the first series, and is the sum of the second.

Theorem 3. If a series converges, then any of its remainders converges. From the convergence of the remainder of the series, the convergence of the series itself follows.

We can say it another way: the convergence of a series is not affected by discarding (or assigning) a finite number of terms in the series. And this property is the most remarkable. Indeed, let the sum of the series be equal to infinity (the series diverges). We add a very large but finite number of terms of the series. This amount can be very large, but, again, it is a finite number. So, this means that the sum of the remainder of the series, and there the members of the series are already negligible numbers, is still equal to infinity due to the infinity of the number of terms.

Theorem 4. A necessary sign of convergence.

If a series converges, then its common term a n tends to zero, i.e. .

Proof. Really,

And if the series converges, then and , and therefore for .

Note that this sign is not sufficient, i.e. the series may diverge, and its common term tends to zero. In example 2, the series diverges, although its common term is .

But if a n does not tend to zero at , then the series is divergent ( a sufficient indication of the divergence of a series).

Convergence of series with positive terms

A series is said to be positive if all .

Partial sums of such a series S n form an increasing sequence, since each previous one is less than the next one, i.e. . From the theory of limits it is known (Bolzano-Weierstrass theorem) that if an increasing sequence is bounded from above (i.e. for all S n there is such a number M, What S n < M for everyone n), then it has a limit. This implies the following theorem.

Theorem. A series with positive terms converges if its partial sums are bounded above, and diverges otherwise.

All are based on this property sufficient signs of convergence of series with positive terms. Let's look at the main ones.

Comparison sign

Let's consider two series with non-negative terms: - (3) and - (4), and starting from some n. Then from the convergence of series (4) the convergence of series (3) follows. And from the divergence of series (3) follows the divergence of series (4).

Otherwise: if a series with larger terms converges, then the series with smaller terms also converges; if a series with smaller terms diverges, then the series with larger terms also diverges.

Example. Examine the series for convergence.

Solution. The general term of a series, and a series is an infinite sum of terms of a geometric progression with a denominator< 1, т.е. это сходящийся ряд. По признаку сравнения (т.к. сходится ряд с б?льшими членами, то сходится и ряд с меньшими) данный ряд сходится.

Comparison sign in extreme form

Consider two series and , and let , be a finite number. Then both series converge or diverge simultaneously.

Example.

Solution. Let's choose a series for comparison, finding out how the general term of the series behaves for large n:

Those. ~ , and as a comparison series we take the series that diverges, as was shown earlier.

Let's calculate the limit

and this means that both rows behave the same way, i.e. this series also diverges.

D'Alembert's sign

Let a series be given and a limit exist. Then if l < 1, то ряд сходится, если l> 1, then the series diverges if l= 1, then this sign does not give an answer (i.e. additional research is necessary).

Example. Examine the series for convergence (recall that, i.e. n-factorial is the product of all integers from 1 to n).

Solution. For this series, (to find it is necessary in instead n substitute n+ 1). Let's calculate the limit

and since the limit is less than 1, this series converges.

Radical Cauchy's sign

Let a series be given and a limit exist. If l< 1, то ряд сходится, если l> 1, then the series diverges if l= 1, then this sign does not give an answer (additional research is necessary).

Example. Examine the series for convergence

Solution. Common member of the series. Let's calculate the limit. This means the series converges.

Integral Cauchy test

Let's consider the series and assume that on the interval XО there is a continuous, positive and monotonically decreasing function such that , n= 1, 2, 3… . Then the series and the improper integral converge or diverge simultaneously.

Note that if a series is given, then the function is considered on the interval.

Let us recall that the indicated improper integral is called convergent if there is a finite limit, and then =. If at does not have a finite limit, then they say that improper integral diverges

Example. Let's consider the series - generalized harmonic series or Dirichlet series with exponent s. If s= 1, then the series is called harmonic series.

We examine this series using the integral Cauchy test: =, and the function = has all the properties specified in the test. Let's calculate the improper integral.

Three cases are possible:

1) s < 1, и тогда

the integral diverges.

2) when s = 1

the integral diverges.

3) if s> 1, then

the integral converges.

Conclusion. The generalized harmonic series converges if s> 1, and diverges if s ≤ 1.

This series is often used for comparison with other series containing degrees n.

Example. Examine the series for convergence.

Solution. For this series ~ =, which means we compare this series with the series, which converges like a Dirichlet series with an exponent s = 2 > 1.

Using the comparison criterion in limiting form, we find the limit of the ratio of the common terms of this series and the Dirichlet series:

Therefore, this series also converges.

Recommendations for usesigns of convergence

First of all, you should use the necessary criterion for the convergence of a series and calculate the limit of the common term of the series at . If , then the series obviously diverges, and if , then one of the sufficient signs should be used.

Signs of comparison It is useful to use in cases where, by transforming the expression for the general term of the series, it is possible to move from the original series to a series whose convergence (or divergence) is known. In particular, if it contains only powers n and does not contain any other functions, it can always be done.

Signs of comparison are used when the original series can be compared with a generalized harmonic series or a series composed of terms of an infinite geometric progression.< применяют, если при замене n . Самой медленно растущей функцией является логарифм, а быстрее всего растёт степенно-показательная функция . Между ними другие известные функции располагаются в следующем порядке:

Therefore, if the numerator contains one of these functions, and the denominator contains a function to the left of it, then most likely the series diverges, and vice versa.

1. If a 1 + a 2 + a 3 +…+a n +…= converges, then the series a m+1 +a m+2 +a m+3 +…, obtained from this series by discarding the first m terms, also converges. This resulting series is called the mth remainder of the series. And, vice versa: from the convergence of the mth remainder of the series, the convergence of this series follows. Those. The convergence and divergence of a series is not violated if a finite number of its terms are added or discarded.

2 . If the series a 1 + a 2 + a 3 +... converges and its sum is equal to S, then the series Ca 1 + Ca 2 +..., where C = also converges and its sum is equal to CS.

3. If the series a 1 +a 2 +... and b 1 +b 2 +... converge and their sums are equal to S1 and S2, respectively, then the series (a 1 +b 1)+(a 2 +b 2)+(a 3 +b 3)+… and (a 1 -b 1)+(a 2 -b 2)+(a 3 -b 3)+… also converge. Their sums are respectively equal to S1+S2 and S1-S2.

4. A). If a series converges, then its nth term tends to 0 as n increases indefinitely (the converse is not true).

- necessary sign (condition)convergence row.

b). If
then the series is divergent - sufficient conditiondivergences row.

-series of this type are studied only according to property 4. This divergent rows.

Sign-positive series.

Signs of convergence and divergence of positive-sign series.

Positive series are series in which all terms are positive. We will consider these signs of convergence and divergence for series with positive signs.

1. The first sign of comparison.

Let two positive-sign series a 1 + a 2 + a 3 +…+a n +…= be given (1) иb 1 +b 2 +b 3 +…+b n +…= (2).

If the members of the series (1) no moreb n and series (2) converges, then series (1) also converges.

If the members of the series (1) no less corresponding members of series (2), i.e. a n b n and row (2) diverges, then series (1) also diverges.

This comparison criterion is valid if the inequality is not satisfied for all n, but only starting from some.

2. Second sign of comparison.

If there is a finite and non-zero limit
, then both series converge or diverge simultaneously.

- rows of this type diverge according to the second criterion of comparison. They must be compared with the harmonic series.

3. D'Alembert's sign.

If for a positive series (a 1 + a 2 + a 3 +…+a n +…= ) exists
(1), then the series converges if q<1, расходится, если q>

4. Cauchy's sign is radical.

If there is a limit for a positive series
(2), then the series converges ifq<1, расходится, если q>1. If q=1 then the question remains open.

5. Cauchy's test is integral.

Let us recall improper integrals.

If there is a limit
. This is an improper integral and is denoted
.

If this limit is finite, then the improper integral is said to converge. The series, respectively, converges or diverges.

Let the series a 1 + a 2 + a 3 +…+a n +…= - positive series.

Let us denote a n =f(x) and consider the function f(x). If f(x) is a positive, monotonically decreasing and continuous function, then if the improper integral converges, then the given series converges. And vice versa: if the improper integral diverges, then the series diverges.

If the series is finite, then it converges.

Rows are very common
-Derichlet series. It converges if p>1, diverges p<1. Гармонический ряд является рядом Дерихле при р=1. Сходимость и расходимость данного ряда легко доказать с помощью интегрального признака Коши.

Basic definitions

Definition. The sum of the terms of an infinite number sequence is called a number series.

In this case, we will call the numbers members of the series, and un - the common term of the series.

Definition. Sums, n = 1, 2, ... are called private (partial) sums of the series.

Thus, it is possible to consider sequences of partial sums of the series S1, S2, …, Sn, …

Definition. A series is called convergent if the sequence of its partial sums converges. The sum of a convergent series is the limit of the sequence of its partial sums.

Definition. If the sequence of partial sums of a series diverges, i.e. has no limit, or has an infinite limit, then the series is called divergent and no sum is assigned to it.

Row Properties

1) The convergence or divergence of the series will not be violated if a finite number of terms of the series are changed, discarded or added.

2) Consider two series and, where C is a constant number.

Theorem. If a series converges and its sum is equal to S, then the series also converges and its sum is equal to CS. (C 0)

3) Consider two rows and. The sum or difference of these series will be called a series where the elements are obtained as a result of the addition (subtraction) of the original elements with the same numbers.

Theorem. If the series and converge and their sums are equal to S and, respectively, then the series also converges and its sum is equal to S +.

The difference of two convergent series will also be a convergent series.

The sum of a convergent and a divergent series is a divergent series.

It is impossible to make a general statement about the sum of two divergent series.

When studying series, they mainly solve two problems: studying convergence and finding the sum of the series.

Cauchy criterion.

(necessary and sufficient conditions for the convergence of the series)

In order for the sequence to be convergent, it is necessary and sufficient that for any there exists a number N such that for n > N and any p > 0, where p is an integer, the inequality would hold:

Proof. (necessity)

Let then for any number there is a number N such that the inequality

is fulfilled when n>N. For n>N and any integer p>0 the inequality also holds. Taking into account both inequalities, we obtain:

The need has been proven. We will not consider the proof of sufficiency.

Let us formulate the Cauchy criterion for the series.

In order for a series to be convergent, it is necessary and sufficient that for any there exist a number N such that for n>N and any p>0 the inequality would hold

However, in practice, using the Cauchy criterion directly is not very convenient. Therefore, as a rule, simpler convergence tests are used:

1) If the series converges, then it is necessary that the common term un tends to zero. However, this condition is not sufficient. We can only say that if the common term does not tend to zero, then the series definitely diverges. For example, the so-called harmonic series is divergent, although its common term tends to zero.

Example. Investigate the convergence of the series

• - the necessary criterion for convergence is not satisfied, which means the series diverges.
• 2) If a series converges, then the sequence of its partial sums is bounded.

However, this sign is also not sufficient.

For example, the series 1-1+1-1+1-1+ … +(-1)n+1+… diverges because the sequence of its partial sums diverges due to the fact that

However, the sequence of partial sums is limited, because for any n.