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Sseries

This U1, U2, ..., UN, ... the function connected to the number of adding numbers in order. The abbreviation of the number level. Such as: U1 U2 ... UN ..., the abbreviation is called the general item of the level, and the part is referred to as the section of the level. If when m → 数, the number of SM has the limit S, then the number of levels convergence, and the sum of s as the harmony, otherwise it is said that the number of levels diverge. The level is an important tool for studying functions. In theory and practical applications, they are in an important position. This is because on the one hand, it can use the number of levels to represent many commonly used non -primaries, and the solution of differential equations is commonly used. On the other hand, the function table can be used as the level, so as to use the level of the level to study the function, such as studying non -primary functions with power levels, and approximate calculations. The convergence of the level is the basic issue of the level theory. From the concept of convergence at the level, it can be seen that the scattered nature of the number of levels is defined by the scattered nature of the part and the number of SM. Therefore, the Cosite Standards that can converge from the number of convergence can obtain the level of convergence of the Caspies: Convergence arbitrarily given positive number ε, which must have natural numbers n. When n ＞ N, all natural numbers P, there are | UN 1 UN 2 … UN P ｜ u003Cε, that is, the absolute value of any paragraph that is fully reached can be small.

. If each UN ≥ 0 (or un≤0) is called the number of positive (or negative) items, the number of positive items and negative items is collectively referred to as the same number. The requirements for the convergence of the positive class level are part of the upper bounds and the sequence SM, such as convergence, because there are infinite number of positives, and the number of infinite numbers is called a variable number. The easiest of which is The level of the shape is called the staggered level. The basic method of judging such a number of convergence is Leibniz's judgment method: if Un ≥UN 1 is established, and each N∈n is established, and the staggered number convergence. For example,

convergence. If the general change number is converged, the number of changes is absolutely convergence. If there is only convergence, but divergent, it is called the number of changing number conditions to converge. For example, absolutely convergence, but only the conditions convergence.

If each level of the number depends on variable X, x changes in a certain interval I, that is, UN = UN (x), X∈i, which is called the number of functional paragraphs, referred to as the function for short, referred to as the function function, for short function, referred to as the function function, for short functions, referred to as function functions, for short functions. series. If X = X0 convergence several levels, it is called X0 convergence point, and a collection composed of convergence points is called convergence domain. If each X∈i is converged, I will be called the convergence range. Obviously, the number of function levels defines a function in its convergence domain, which is called a function S (x), that is, if you meet stronger conditions, consensus in S (x) in the convergence domain.

The important function level number is the level of the shape, which is called the power level. Its structure is simple, converging domain is a interval (not necessarily included), and has a similar polynomial properties in a certain range. It can perform calculations such as differentials and points in the convergence range. For example, the convergence range of the power level is that the convergence range of the power level is [1, 3], and the power level converges on the real number.

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Mathematical analysis (Analysis) is a mathematical major One of the compulsory courses is that the basic content is calculus, but it is very different from the calculus.

The compliance is a collective name of Calculus and Integral Caculus. This is because early micro -points are mainly used for calculation problems in astronomical, mechanics, and geometry. Later, people also referred to micro -accumulation as an analysis (Analysis) or infinite analysis, which specifically referred to the science of calculating the calculation of calculation and calculation problems such as the use of infinite or infinite or infinite.

The early calculus, because it cannot make a convincing explanation of the infinite concepts, it will not be developed for a long time. Cauchy and the later Welstras () laid a solid theoretical foundation for the calculus, and the calculus gradually evolved into a very strict mathematical discipline, known as "mathematical analysis".

The basis of mathematical analysis is real theory. The most important characteristic of the real number system is continuity. With the continuity of real numbers, we can discuss limits, continuity, micro -scores and points. It is in the process of discussing the legitimacy of various extreme operations of the function that people have gradually established a strict mathematical analysis theoretical system.

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R n is a micro -score, points, and mathematical branches of the study function and mathematical branches related to concepts and applications. Calculus points are based on real numbers, functions, and limits.

The concept of limit and calculus can be traced back to ancient times. In the second half of the seventeenth century, Newton and Leibnitz completed many mathematicians who had participated in preparations and established microstics independently. The starting point of the establishment of calculus is an intuitive infinite amount, and the theoretical foundation is not firm. It wasn't until the nineteenth century that Kosi and Welstras established extreme theory. Condor and others established a strict real -number theory that this discipline was tightly eclipsed.

The calculus is developed with actual applications. It is in the branch of natural sciences, social sciences and application sciences in astronomy, mechanics, chemistry, biology, engineering, economics, etc. More and more widely used. In particular, the invention of the computer is more helpful for the continuous development of these applications.

The components are the general term for micro -scores and points.

The things in the objective world, small to particles, to the universe, are always exercising and changing. Therefore, after introducing the concept of variables in mathematics, math phenomena may be described as mathematics.

The deepening of functional concepts and deepening of the function of function, and due to the needs of science and technology development, a new mathematical branch arises after analysis of geometry. This is calculus. The status of the discipline of micro -accumulation in mathematics is very important. It can be said that it is the largest creation in all mathematics after Eushi geometry.

The establishment of micro -integral science

It from micro -accumulation to a discipline, it was in the seventeenth century, but the idea of ​​micro -scores and points had been generated in ancient times. Essence

In AD in the third century BC, the problems of the ancient Greek Akimids in the problem of solving the area, ball and ball crown area, threaded area and rotating bisodes of the bioplasm. It implies the idea of ​​modern points. As the limit theory of the foundation of micromics, it was a clear discussion in ancient times. For example, in the book "Zhuangzi" written by Zhuang Zhou in my country, it is recorded that "one foot, take half of the day, and endlessly." During the Three Kingdoms period, Liu Hui mentioned in his cut operation, "Simply, Lost Small, Cutting and Cutting, so that it is not cut, but there is nothing to lose with the circular and body." These are all these. "These are all these." Simple and typical extreme concepts.

In the 17th century, many scientific issues need to be solved, and these problems have become factor that promotes micro -accumulation. In the end, there are about four main types of problems: the first category is to directly appear during the study of the movement, that is, the problem of seeking instant speed. The second type of problem is the problem of cutting the curve. The third type of problem is the maximum and minimum value of the function. The fourth type of problem is the area of ​​long curve, the size of the curve, the volume of the curve, the ingenuity of the object, the center of gravity of the object, and the gravity of a relatively large object on the other object.

many famous mathematicians, astronomers, and physicists in the seventeenth century have done a lot of research work to solve the above -mentioned problems, such as French, Descartes, Robber, Wattimonia, Tip Sha; British Baro, Varis; Cairpler in Germany; Cavaleli and others in Italy have proposed many very good theories. Contributions to the establishment of calculus.

The second half of the seventeenth century, on the basis of the work of predecessors, British scientists ㄈ Newton and German mathematician Leibniz separately studied and completed the calculus points in their own kingdoms. Founding work, although this is just a very preliminary job. Their maximum achievements are to connect the two seemingly unrelated issues, one is the threading problem (the central issue of the micro -science), and the other is the problem of accumulation (the central issue of points).

The starting point for the establishment of calculus in Newton and Leibniz is an intuitive infinite amount, so this discipline was also called infinite small analysis in the early stage. source. Newton's studies focused on the consideration of sports, but Leibnitz focused on geometry.

Newton wrote the "Flowing Method and Infinite Number" in 1671. This book was published until 1736. It pointed out in this book that the variables are from dots, lines, and faces. The continuous exercise deny the static collection of the variables that I think before. He calls continuous variables as flow volume and the number of flows of these flows is called flow. The central problem proposed by Newton's flowing operation is: the path of known continuous exercise, the speed of the given time (micro -division method); the speed of the known movement is required to go through the journey (points method) through the given time.

Telabnitz in Germany is a scholar of talents. In 1684, he published the earliest micro -accumulation literature in the world. This article has a long and weird The name "A new method of seeking extremely small and very small and cutting lines, it is also suitable for division and unreasonable amounts, and the wonderful types of this new method". This is such an article that is quite reasonable and vague, but it has the epoch -making significance. He contains modern micro -symbols and basic differential rules. In 1686, Leibnitz published the first science literature. He is one of the greatest symbols in history. The calculus symbol he created is far better than Newton's symbol, which has a great impact on the development of calculus. The calculus universal symbols we use now were carefully selected by Leibniz at the time.

The establishment of micro -points has greatly promoted the development of mathematics. In the past, many primary mathematics had no problem. The use of calculus was often solved, showing the extraordinary power of micro -accumulation.

If mentioned earlier that the establishment of a science is not the performance of a person. After how many people have worked hard, on the basis of accumulating a lot of achievements, they will finally be someone or in the end. Several people summarized. The same is true of calculus.

I unfortunate things, because people appreciate the magnificent effects of the calculus, when they propose the founder of this subject, it has caused a big wave, causing the European continental continent The long -term confrontation of mathematicians and British mathematicians. British mathematics closed the country during a period of time, and was prejudiced in national prejudice. It was too close to Newton's "flowing skills". Therefore, the development of mathematics was a hundred years behind.

In fact, Newton and Leibnitz are independent research, which are completed in a general similar time. The special thing is that Newton's founding of calculus is about 10 years earlier than Leibuni, but it is the theory of publishing calculus publicly, but Leibniz has published three years earlier than Newton. Their research has their own strengths and their own weaknesses. At that time, due to national prejudice, the debate on the priority of invention continued for more than 100 years since 1699.

The should be pointed out that this is the same as the completion of any major theory in history for a period of time. The work of Newton and Leibniz is also very imperfect. They are different and vague on the issue of infinity and infinity. Newton's infinite minimum, sometimes zero, sometimes not zero but limited small amount; Leibnitz's cannot be expressed by himself. These foundation defects eventually led to the occurrence of the second math crisis.

Until the beginning of the 19th century, scientists at the French School of Sciences were headed by Cosite, carefully studied the theory of calculus, and established extreme theory. Later The strictness of the limit has made the theory a firm foundation for calculus. Only then did the calculus develop further.

Any emerging, infinitely promising scientific achievement attracts the majority of scientific workers. There are some stars in the history of calculus: Switzerland's Yakob Betunley and his brother John Benuly, Euler, French Lagram, Cosite ...

Or's geometry, ancient and medieval generation mathematics, are a constant mathematics, calculus is the real variable mathematics, and the great revolution in mathematics. Calculus points are the main branches of higher mathematics, not just limited to solving the problem of changes in mechanics. It has established countless great achievements in modern and modern science and technology gardens.

The basic content of the calculus

The research function. Studying the movement of things from the perspective of quantity is the basic method of calculus. This method is called mathematical analysis.

It, in a broad sense, mathematical analysis includes many branch disciplines such as calculus and function theory, but now it is generally used to equating mathematical analysis with calculus. Mathematical analysis has become synonymous words of calculus points. As soon as the mathematical analysis, you know that the calculus refers to the calculus. The basic concepts and contents of calculus include micro -science and integration.

The main contents of micro -science include: extreme theory, guide number, differential division, etc.

The main contents of integral science include: points, irregular points, etc.

The calculus was developed with the application. At first, the Microchlery and Micro -division equations of Newton used to export the three laws of the Kaipus Planet Movement from the law of gravity. Since then, micro -accumulation has greatly promoted the development of mathematics, and at the same time, it has also greatly promoted natural sciences, social sciences, and social sciences and application sciences in astronomy, mechanics, physics, chemistry, biology, engineering, economics, etc. development of. And in these disciplines, it is increasingly widely used, especially the emergence of computers is more helpful for the continuous development of these applications.

r

definitions

setting function y = f (x) is defined in a certain range, x0 and x0 Δx in this interval Inside. If the function of the function Δy = f (x0 Δx) − f (x0) can be represented as Δy = a Δx0 O (Δx0) (where A is not dependent on the constant of Δx), and O (Δx0) is higher than Δx higher than Δx. The order of the order is infinite, so it is called the function f (x) at the point X0 that is slightly slightly, and A ΔX is called a differential point of the function at the point X0 corresponding to the independent variable increment Δx, which is recorded as DY, that is, DY = AΔX.

usually referred to the incremental Δx of the independent variable X as the differential of the independent variable, which is DX, that is, dx = Δx. So the differential of the function y = f (x) can be recorded as dy = f (x) dx. The micro -division of the function and the differential quotient of the independent variable are equal to the number of the function of the function. Therefore, the guide is also called Weishang.

The geometric significance

The increase in the point m on the point m on the curve y = f (x) on the horizontal coordinates. In the increase in the vertical marking, the DY is the incremental increase of Δx on the vertical marker of the cut line of the curve. When | Δx | very hours, | Δy -dy | Δy | much smaller (high -level infinitely small), so near the point M, we can use the cut segment to approximate the curve segment.

D diversity

. When the independent variable is multiple, the diversified micro -score can be defined.

The points are slightly differential calculations, that is, the guide function of the function knows the original function. In terms of application, the effect of points is not only that. It is used in large quantities to seek peace. It is popular to find the area of ​​the triangle edge. This clever solution is determined by the special nature of integration.

The uncertain integration (also known as the original function) of a function refers to another family function. The guide function of this family function is exactly the previous function.

where: [f (x) c] = f (x)

Essence It is equal to the value of a function of the function at the value of A at the value of B.

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n theory

The mathematical branch of random phenomenon quantity. Random phenomena are relative to decisive phenomena. Under certain conditions, the phenomenon of a certain result is called a decisive phenomenon. For example, under the standard atmosphere, the water will inevitably boil when the pure water is heated to 100 ° C. Random phenomenon refers to a series of tests or observations that will get different results when the basic conditions are unchanged. Before each test or observation, there will be no result, and it will appear accidental. For example, if you throw a coin, you may appear on the front or opposite side, and the light bulbs produced under the same process, the long -term ginseng is not uniform, and so on. The realization of random phenomena and observation of it are called random tests. Each possible result of the random test is called a basic event, and one or one basic event collectively refers to random events, or the incident for short. The probability of the event is to measure the possibility of the incident. Although the occurrence of an event in a random test is accidental, those random trials that can be repeated under the same conditions often show obvious quantitative laws. For example, a uniform coin continuously throwing a continuous number of times, the frequency of the front is gradually tended to 1/2 with the increase of the number of throws. For another example, the length of a one object is measured multiple times. The average value of the measurement results gradually stabilizes the number of constants with the number of measurement times, and the measured values ​​are mostly near this constant. Symmetry of less and certain levels. Large laws and central limitations are described and demonstrated these laws. In actual life, people often need to study the random process of the evolution of a certain random phenomenon. For example, tiny particles are randomly collided by the surrounding molecules in the liquid to form irregular movements (that is, Brown Movement), which is the random process. The probability of statistical characteristics and calculations related to some events related to the random process, especially the problems related to the research on the random process sample track (that is, one realization of the process), are the main topics of modern probability theory. Probability theory is closely related to actual life. It is widely used in natural sciences, technical sciences, social sciences, military and industrial and agricultural production.

The origin of probability theory is related to gambling issues. In the 16th century, Italian scholars began to study some simple problems in gambling such as dice. In the middle of the 17th century, French mathematician B. Pascar, P.DE Ferma and Dutch mathematician C. Huygos studied some more complex gambling problems based on the arrangement and combination of arrangements. Questions, etc. With the development of science in the 18th and 19th centuries, people have noticed that there is some similarity between certain creatures, physics and social phenomena and opportunity games, so as to be applied to these fields by the probability theory of the origin of the game; at the same time It has also greatly promoted the development of probability theory itself. The founder of the probability theory became a branch of mathematics was a Swiss mathematician J. Berninley. He established the first limited theory in probability theory, that is, the law of Bernuyli, which clarified the frequency of the incident to stabilize it from its stability to its. Probability. Subsequently, A.DE Morf and P.S. Laplas exported the original form of the second basic limit (central limited theory).拉普拉斯在系统总结前人工作的基础上写出了《分析的概率理论》，明确给出了概率的古典定义，并在概率论中引入了更有力的分析工具，将概率论推向A new stage of development. At the end of the 19th century, Russian mathematician P.L. Chebibbi Snow, A.A. Marco, A.M. Leejunov and others used analysis methods to establish a general form of large number of laws and central limited theory. Scientifically explained why many random random encountered in the actual situation actually encountered in the actual situation Variables approach the normal distribution. In the early 20th century, people were stimulated by physics, and people began to study the random process. In this regard, A.N. Cormorov, N. Venin, A.A. Marco, A.R Xinqin, P. Levi and W. Feller and others made outstanding contributions.

How to define the probability and how to build probability theory on the basis of strict logic is the difficulty of the development of probability theory. The exploration of this issue has continued for 3 centuries. The Libeg measurement and integration theory and the abstract measurement and integral theory of the initial development of the Libeg were laid the foundation for the establishment of the probability axiom system. In this context, Soviet mathematician Kormorov first gave the definition of probability measurement theory and a strict axiom system in his book "Basics of Probability" in 1933. His axioms have become the basis of modern probability theory, making probability theory a rigorous mathematical branch, and play a positive role in the rapid development of probability theory.

1. The convergence radius of the power level is 0 to indicate that it is the divergent number
2, the one-dollar function limit is very simple. As long as the number of the axis is judged when the X → X0 (fully close), the absolute value of the Y-Y0 can be arbitrarily arbitrarily arbitrarily arbitrarily arbitrarily. Just small.
The dual function is actually the limit of complex numbers. You may wish to set the reciprocating function w = f (z) to define the domain as D, and on the plane: let the small positive number ε> 0, there is Δ> 0, should Double -digital Z∈ disc nellione U (Z0, Δ) ∩D,
f (z) ∈ ∈ CD -neighboring domain E (A, ε), when Z → Z0, the limit of f (z) is A Essence
It can be seen that the one -dollar function limit only considers one dimension axis, and the dual function should consider the plane (that is, the adjacent domain of the disc)
. There is also a different relationship between different divisions and partial guidance.