For many people, mathematical analysis is just a set of incomprehensible numbers, icons and definitions that are far from real life. However, the world in which we exist is built on numerical patterns, the identification of which helps not only to learn about the world around us and solve its complex problems, but also to simplify everyday practical tasks. What does a mathematician mean when he says that a number sequence converges? This should be discussed in more detail.
What is an infinitesimal?
Let's imagine matryoshka dolls that fit one inside the other. Their sizes, written in the form of numbers, starting with the largest and ending with the smallest of them, form a sequence. If you imagine an infinite number of such bright figures, then the resulting row will be fantastically long. This is a convergent number sequence. And it tends to zero, since the size of each subsequent nesting doll, catastrophically decreasing, gradually turns into nothing. So it's easycan be explained: what is infinitesimal.
A similar example would be a road leading into the distance. And the visual dimensions of the car driving away from the observer along it, gradually shrinking, turn into a shapeless speck resembling a dot. Thus, the car, like an object, moving away in an unknown direction, becomes infinitely small. The parameters of the specified body will never be zero in the truest sense of the word, but invariably tend to this value in the final limit. Therefore, this sequence converges again to zero.
Calculate everything drop by drop
Let's imagine now a worldly situation. The doctor prescribed the patient to take the medicine, starting with ten drops a day and adding two every next day. And so the doctor suggested continuing until the contents of the vial of medicine, the volume of which is 190 drops, run out. It follows from the foregoing that the number of such, scheduled by day, will be the following number series: 10, 12, 14 and so on.
How to find out the time to complete the entire course and the number of members of the sequence? Here, of course, you can count the drops in a primitive way. But it is much easier, given the pattern, to use the formula for the sum of an arithmetic progression with a step d=2. And using this method, find out that the number of members of the number series is 10. In this case, a10=28. The member number indicates the number of days of taking the medicine, and 28 corresponds to the number of drops that the patient shoulduse on the last day. Does this sequence converge? No, because despite the fact that it is limited to 10 from below and 28 from above, such a number series has no limit, unlike the previous examples.
What's the difference?
Let's now try to clarify: when the number series turns out to be a convergent sequence. A definition of this kind, as can be concluded from the above, is directly related to the concept of a finite limit, the presence of which reveals the essence of the issue. So what is the fundamental difference between the previously given examples? And why in the last of them, the number 28 cannot be considered the limit of the number series X =10 + 2(n-1)?
To clarify this question, consider another sequence given by the formula below, where n belongs to the set of natural numbers.
This community of members is a set of common fractions, the numerator of which is 1, and the denominator is constantly increasing: 1, ½ …
Moreover, each successive representative of this series approaches 0 more and more in terms of location on the number line. And this means that such a neighborhood appears where the points cluster around zero, which is the limit. And the closer they are to it, the denser their concentration on the number line becomes. And the distance between them is catastrophically reduced, turning into an infinitesimal one. This is a sign that the sequence is converging.
SimilarThus, the multi-colored rectangles shown in the figure, when moving away in space, are visually more crowded, in the hypothetical limit turning into negligible.
Infinitely large sequences
Having analyzed the definition of a convergent sequence, let's move on to counterexamples. Many of them have been known to man since ancient times. The simplest variants of divergent sequences are the series of natural and even numbers. They are called infinitely large in a different way, since their members, constantly increasing, are increasingly approaching positive infinity.
An example of such can also be any of the arithmetic and geometric progressions with step and denominator, respectively, greater than zero. In addition, numerical series are considered divergent sequences, which do not have a limit at all. For example, X =(-2) -1.
Fibonacci sequence
The practical benefits of the previously mentioned number series for humanity is undeniable. But there are countless other great examples. One of them is the Fibonacci sequence. Each of its terms, which begin with one, is the sum of the previous ones. Its first two representatives are 1 and 1. The third 1+1=2, the fourth 1+2=3, the fifth 2+3=5. Further, according to the same logic, the numbers 8, 13, 21 and so on follow.
This series of numbers increases indefinitely and has nofinal limit. But it has another wonderful property. The ratio of each previous number to the next one is getting closer and closer in its value to 0.618. Here you can understand the difference between a convergent and divergent sequence, because if you make a series of received partial divisions, the indicated numerical system will have a finite limit equal to 0.618.
Sequence of Fibonacci ratios
The number series indicated above is widely used for practical purposes for the technical analysis of markets. But this is not limited to its capabilities, which the Egyptians and Greeks knew and were able to put into practice in ancient times. This is proved by the pyramids they built and the Parthenon. After all, the number 0.618 is a constant coefficient of the golden section, well known in the old days. According to this rule, any arbitrary segment can be divided in such a way that the ratio between its parts will coincide with the ratio between the largest of the segments and the total length.
Let's construct a series of the indicated relations and try to analyze this sequence. The number series will be as follows: 1; 0.5; 0.67; 0.6; 0.625; 0.615; 0, 619 and so on. Continuing in this way, we can make sure that the limit of the convergent sequence will indeed be 0.618. However, it is necessary to note other properties of this regularity. Here the numbers seem to go randomly, and not at all in ascending or descending order. This means that this convergent sequence is not monotone. Why this is so will be discussed further.
Monotonicity and limitation
Members of the number series can clearly decrease with increasing number (if x1>x2>x3 >…>x >…) or increasing (if x1<x2<x 3<…<x <…). In this case, the sequence is said to be strictly monotonic. Other patterns can also be observed, where the numerical series will be non-decreasing and non-increasing (x1≧x2≧x3≧ …≧x ≧… or x1≦x2≦x3 ≦…≦x ≦…), then the successively convergent one is also monotonic, only not in the strict sense. A good example of the first of these options is the number series given by the following formula.
Having painted the numbers of this series, you can see that any of its members, indefinitely approaching 1, will never exceed this value. In this case, the convergent sequence is said to be bounded. This happens whenever there is such a positive number M, which is always greater than any of the terms of the series modulo. If a number series has signs of monotonicity and has a limit, and therefore converges, then it is necessarily endowed with such a property. And the opposite doesn't have to be true. This is evidenced by the boundedness theorem for a convergent sequence.
The application of such observations in practice is very useful. Let's give a specific example by examining the properties of the sequence X =n/n+1, and prove its convergence. It is easy to show that it is monotonic, since (x +1 – x) is a positive number for any n values. The limit of the sequence is equal to the number 1, which means that all the conditions of the above theorem, also called the Weierstrass theorem, are satisfied. The theorem on the boundedness of a convergent sequence states that if it has a limit, then in any case it turns out to be bounded. However, let's take the following example. The number series X =(-1) is bounded from below by -1 and from above by 1. But this sequence is not monotonic, has no limit, and therefore does not converges. That is, the existence of a limit and convergence does not always follow from limitation. For this to work, the lower and upper limits must match, as in the case of Fibonacci ratios.
Numbers and laws of the Universe
The simplest variants of a convergent and divergent sequence are perhaps the numerical series X =n and X =1/n. The first of them is a natural series of numbers. It is, as already mentioned, infinitely large. The second convergent sequence is bounded, and its terms are close to infinitesimal in magnitude. Each of these formulas personifies one of the sides of the multifaceted Universe, helping a person to imagine and calculate something unknowable, inaccessible to limited perception in the language of numbers and signs.
The laws of the universe, ranging from negligible to incredibly large, also expresses the golden ratio of 0.618. Scientiststhey believe that it is the basis of the essence of things and is used by nature to form its parts. The relations between the next and the previous members of the Fibonacci series, which we have already mentioned, do not complete the demonstration of the amazing properties of this unique series. If we consider the quotient of dividing the previous term by the next one through one, then we get a series of 0.5; 0.33; 0.4; 0.375; 0.384; 0.380; 0, 382 and so on. It is interesting that this limited sequence converges, it is not monotonous, but the ratio of the neighboring numbers extreme from a certain member always approximately equals 0.382, which can also be used in architecture, technical analysis and other industries.
There are other interesting coefficients of the Fibonacci series, they all play a special role in nature, and are also used by man for practical purposes. Mathematicians are sure that the Universe develops according to a certain "golden spiral", formed from the indicated coefficients. With their help, it is possible to calculate many phenomena occurring on Earth and in space, from the growth in the number of certain bacteria to the movement of distant comets. As it turns out, the DNA code obeys similar laws.
Declining geometric progression
There is a theorem that asserts the uniqueness of the limit of a convergent sequence. This means that it cannot have two or more limits, which is undoubtedly important for finding its mathematical characteristics.
Let's look at somecases. Any numerical series composed of members of an arithmetic progression is divergent, except for the case with a zero step. The same applies to a geometric progression, the denominator of which is greater than 1. The limits of such numerical series are the “plus” or “minus” of infinity. If the denominator is less than -1, then there is no limit at all. Other options are possible.
Consider the number series given by the formula X =(1/4) -1. At first glance, it is easy to see that this convergent sequence is bounded because it is strictly decreasing and in no way capable of taking negative values.
Let's write a number of its members in a row.
It will turn out: 1; 0.25; 0.0625; 0.015625; 0, 00390625 and so on. Quite simple calculations are enough to understand how quickly this geometric progression decreases from the denominators 0<q<1. While the denominator of the terms increases indefinitely, they themselves become infinitesimal. This means that the limit of the number series is 0. This example once again demonstrates the limited nature of the convergent sequence.
Fundamental sequences
Augustin Louis Cauchy, a French scientist, revealed to the world many works related to mathematical analysis. He gave definitions to such concepts as differential, integral, limit, and continuity. He also studied the basic properties of convergent sequences. In order to understand the essence of his ideas,some important details need to be summarized.
At the very beginning of the article, it was shown that there are such sequences for which there is a neighborhood where the points representing the members of a certain series on the real line begin to cluster, lining up more and more densely. At the same time, the distance between them decreases as the number of the next representative increases, turning into an infinitely small one. Thus, it turns out that in a given neighborhood an infinite number of representatives of a given series are grouped, while outside of it there are a finite number of them. Such sequences are called fundamental.
The famous Cauchy criterion, created by a French mathematician, clearly indicates that the presence of such a property is sufficient to prove that the sequence converges. The reverse is also true.
It should be noted that this conclusion of the French mathematician is mostly of purely theoretical interest. Its application in practice is considered to be a rather complicated matter, therefore, in order to clarify the convergence of series, it is much more important to prove the existence of a finite limit for a sequence. Otherwise, it is considered divergent.
When solving problems, one should also take into account the basic properties of convergent sequences. They are shown below.
Infinite sums
Such famous scientists of antiquity as Archimedes, Euclid, Eudoxus used the sums of infinite number series to calculate the lengths of curves, volumes of bodiesand areas of figures. In particular, in this way it was possible to find out the area of the parabolic segment. For this, the sum of the numerical series of a geometric progression with q=1/4 was used. The volumes and areas of other arbitrary figures were found in a similar way. This option was called the "exhaustion" method. The idea was that the studied body, complex in shape, was broken into parts, which were figures with easily measured parameters. For this reason, it was not difficult to calculate their areas and volumes, and then they added up.
By the way, similar tasks are very familiar to modern schoolchildren and are found in USE tasks. The unique method, found by distant ancestors, is by far the simplest solution. Even if there are only two or three parts into which the numerical figure is divided, the addition of their areas is still the sum of the number series.
Much later than the ancient Greek scientists Leibniz and Newton, based on the experience of their wise predecessors, learned the patterns of integral calculation. Knowledge of the properties of sequences helped them solve differential and algebraic equations. At present, the theory of series, created by the efforts of many generations of talented scientists, gives a chance to solve a huge number of mathematical and practical problems. And the study of numerical sequences has been the main problem solved by mathematical analysis since its inception.
