Mathematics is the science that builds the world. As a scientist, and a simple person - no one can do without it. First, small children are taught to count, then add, subtract, multiply and divide, secondary symbols are used for secondary school, and in the older one they can not do without them.
But today we will talk about what all the known mathematics are built on. About the community of numbers called "limits of sequences".
What are sequences and where is their limit?
The meaning of the word "sequence" is not difficult to interpret. It is the construction of things where someone or something is located in a certain order or line. For example, the queue for tickets to the zoo is a sequence. And it can only be one! If, for example, look at the queue in the store - this is one sequence. And if one person from this queue suddenly leaves, then this is another line, another order.
The word "limit" is also easily interpreted - this is the end of something. However, in mathematics, the limits of sequences are those on the number line to which the sequence of numbers tends. Why does it seek, not end? It's simple, the number line has no end, and most of the sequences, like rays, have only a beginning and look like this:
Hence the definition of a sequence is a function of the natural argument. In simpler words, this is the series of terms of a certain set.
How is the numerical sequence constructed?
The simplest example of a numerical sequence can look like this: 1, 2, 3, 4, ... n ...
In most cases, for practical purposes, sequences are constructed from numbers, and each subsequent member of the series, denoted by X, has a name. For example:
x1 - the first member of the sequence;
x2 The second term of the sequence;
x3 - the third term;
xn - Enne member.
In practical methods, a sequence is given by a general formula in which there is some variable. For example:
Xn = 3n, then the series of numbers will look like this:
It is worth remembering that for general recording of sequences you can use any Latin letters, 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 notion of such a numerical series, which all faced, being in middle classes. Arithmetic progression is a series of numbers in which the difference between neighboring members is constant.
The problem: "Let a1 = 15, and the step of progression of the numerical series d = 4. Build the first 4 members of this series "
Solution: a1 = 15 (by condition) - the first member of the progression (numerical series).
a2 = 15 + 4 = 19 - the second member of the progression.
a3 = 19 + 4 = 23 - the third term.
a4 = 23 + 4 = 27 is the fourth term.
Most of the sequences are infinite, it's worth remembering for life. There are two interesting types of numerical series. The first is given by the formula an = (-1) n. Mathematicians often call this sequence a flasher. Why? Let us check its number series.
-1, 1, -1. 1, -1, 1, etc. On such an example it becomes clear that the numbers in the sequences can easily be repeated.
The factorial sequence. It is easy to guess - in the formula defining the sequence, there is a factorial. For example: an = (n + 1)!
Then the sequence will look like this:
a3 = 1x2x3x4 = 24, and so on.
There is even a sequence consisting of the same number. Thus, an = 6 consists of an infinite set of sixes.
Determination of the limit
Limits of sequences have long existed in mathematics. Of course, they deserve their own competent design. So, it's time to learn the definition of the limits of sequences. To begin with, consider in detail the limit for a linear function:
- All limits are abbreviated lim.
- A limit entry consists of a shortening of lim, some variable tending to a certain number, zero or infinity, and also from 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 number to which all the terms of the sequence approach infinitely. A simple example: ax = 4x + 1. Then the sequence itself will look like this.
5, 9, 13, 17, 21 ... x ...
Thus, the given sequence will increase infinitely, and, therefore, its limit is equal to infinity as x → ∞, and it should be written like this:
If we take a similar sequence, but x will tend to 1, we get:
A number of numbers will be: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number more and more to the unit (0.1, 0.2, 0.9, 0.986). From this series it is clear that the limit of a function is five.
From this part it is worth remembering what is the limit of a numerical sequence, the definition and method of solving simple tasks.
General designation of the limit of sequences
Having analyzed the limit of a numerical sequence, its definition and examples, it is possible to start a more complex topic. Absolutely all the limits of sequences can be formulated with a single formula, which is usually disassembled in the first semester.
So, what does this set of letters, modules and signs of inequalities mean?
∀ is the universal quantifier replacing the phrases "for all", "for everything", etc.
∃ - existence quantifier, in this case means that there is a certain value of N, which belongs 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," and so on.
Next comes the module. Obviously, the module is the distance, which by definition can not be negative. Hence the modulus of the difference is strictly less than the "epsilon."
To fix the material, read the formula aloud.
Uncertainty and definiteness of the limit
The method of finding the limit of sequences, which was considered above, even though simple in application, but not so rational in practice. Try to find the limit for this function:
If we substitute the various values of "X" (each time increasing: 10, 100, 1000, etc.), then we get ∞ in the numerator, but ∞ in the denominator. It turns out a rather strange fraction:But is it really? To calculate the limit of a numerical sequence in this case, it seems easy enough. You could leave everything as is, because the answer is ready, and received it on reasonable terms, however, there is another method specifically for such cases.
To begin with, we find the highest degree in the numerator of the fraction - this is 1, since x can be represented as x 1.
Now we find the highest degree in the denominator. Also 1.
We divide both the numerator and the denominator into a variable in the highest degree. In this case, we divide the fraction by x 1.
Next, we find to what value each term that contains the variable tends. In this case, fractions are considered. As x → ∞, the value of each of the fractions tends to zero. When writing in writing, it is worth making such 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 completely allowed not to take it into account in the calculations. In fact, x will never be 0 in this case, because you can not divide by zero.
What is the neighborhood?
Suppose, at the disposal of the professor, a complex sequence, presumably given by an equally complex formula. The professor found the answer, but is it suitable? After all, all people are wrong.
Auguste Cauchy in his time came up with a great way to prove the limits of sequences. His method was called operating with neighborhoods.
Suppose that there is a point a, its neighborhood in both directions on the number line is equal to ε ("epsilon"). Since the last variable is distance, its value is always positive.
Now we give a sequence xn and assume that the tenth member of the sequence (x10 ) occurs in a neighborhood of a. How to write this fact in mathematical language?
Assume that x10 is to the right of a, then the distance x10 -a \u0026 lt; ε, however, if "tenth" is located to the left of the point a, then the distance turns out to be negative, and this is impossible, therefore, the left side of the inequality must be added to the module. It turns out that | x10 -a | \u0026 lt; ε.
Now it's time to clarify in practice the formula, which was mentioned above. A certain number a is said to be a finite point of the sequence if for any of its limits the inequality ε\u003e 0 holds, and the entire neighborhood has its natural number N such that all terms of the sequence with more significant numbers appear inside the sequence | xn - a | \u0026 lt; ε.
With such knowledge it is easy to implement the decision of the limits of the sequence, to prove or disprove the ready answer.
Theorems on the limits of sequences are an important part of the theory, without which practice is impossible. There are only four main theorems, remembering which, one can simplify the course of the decision or evidence at times:
- Uniqueness of the limit of a sequence. The limit for any sequence can be only one or not at all. The same example with a queue that can have only one end.
- If the series of numbers has a limit, then the sequence of these numbers is bounded.
- The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
- The limit of the quotient of the division of two sequences is equal to the quotient limit if and only if the denominator does not vanish.
Proof of Sequences
Sometimes it is required to solve the inverse problem, to prove the given limit of a numerical sequence. Consider the example.
Prove that the limit of the sequence given by the formula is zero.
By the rule considered above, for any sequence the inequality | xn - a | \u0026 lt; ε. We substitute the given value and the reference point. We get:
We express n through "epsilon" to show the existence of a certain number and to prove the existence of a limit of sequence.
At this stage, it is important to recall that "epsilon" and "en" are positive numbers and not equal to zero. Now we can continue the further transformations, using knowledge about the inequalities obtained in the secondary school.
Here such a convenient method can prove the limit of a numerical sequence, no matter how complex it may seem at first glance. The main thing is not to panic when seeing the task.
Maybe it's not there?
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" xn = (-1) n. it is obvious that a sequence consisting of only two digits, cyclically repeating, can not have a limit.
The same story is repeated with sequences consisting of one number, fractional, having in the course of computations an uncertainty of any order (0/0, ∞ / ∞, ∞ / 0, etc.). However, it should be remembered that an incorrect calculation also takes place. Sometimes the limit of the followers will be found by rechecking your own decision.
The monotone sequence
Above we considered several examples of sequences, methods for solving them, and now we try to take a more specific case and call it a "monotonous sequence".
Definition: any sequence is validly called monotonically increasing if it satisfies the strict inequality xn \u0026 lt; xn+1. Also, any sequence is said to be monotone decreasing if it satisfies the inequality xn \u0026 gt; xn+1.
Along with these two conditions, there are also similar non-strict inequalities. Correspondingly, xn ≤ xn+1 (nondecreasing sequence) and xn ≥ xn+1 (nonincreasing sequence).
But it is easier to understand this by examples.
The sequence given by the formula xn = 2 + n, forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.
Limit of convergent and bounded sequence
A bounded sequence is a sequence having a limit. A convergent sequence is a series of numbers having an infinitesimal limit.
Thus, the limit of a bounded sequence is any real or complex number. Remember that there can be only one limit.
The limit of a convergent sequence is an infinitesimal (real or complex) value. If you draw a sequence diagram, then at a certain point it will seem to converge, strive to address a certain amount. Hence the name is a convergent sequence.
The limit of a monotone sequence
The limit of such a sequence may or may not be. At first it is useful to understand when it is, hence it is possible to pounce on the proof of the absence of a limit.
Among the monotonous sequences, the convergent and divergent are distinguished. Convergent is a sequence that is formed by the set x and has a real or complex limit in the given set. Divergent is a sequence that does not have a limit in its set (neither real nor complex).
And the sequence converges, if the geometric image of its upper and lower limits converge.
The limit of a convergent sequence in many cases can be equal to zero, since any infinitesimal sequence has a known limit (zero).
What a convergent sequence they take, they are all limited, but not all limited sequences converge.
The sum, the difference, the product of two convergent sequences is also a convergent sequence. However, the quotient can also be convergent if it is defined!
Various actions with limits
Sequence limits are the same essential (in most cases) value, as well as numbers and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits.
First, like numbers, limits of all sequences you can add and subtract. On the basis of the third theorem about limits of sequences, fairly following equality: limit of the sum of the sequence equal to the sum of their limits.
Secondly, based on the fourth theorem on the limits of sequences, the following equality holds: the limit of the product of the n-th number of sequences is equal to the product of their limits. The same is true for division: the limit of a particular two sequences is equal to their particular limits, provided that the limit is not zero. After all, if the limit of sequences is zero, then we get a division by zero, which is impossible.
Properties of Sequence Values
- The sum of any number of arbitrarily small quantities will also be small.
- The sum of any number of large quantities will be infinitely large.
- The product of arbitrarily small quantities is infinitesimal.
- The product of arbitrarily large numbers is infinitely large.
- If the original sequence tends to an infinitely large number, then the inverse of it will be infinitesimal and tend to zero.
In fact, computing 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 requiring maximum attention and perseverance. Of course, it's enough just to grasp the essence of the solution of such expressions. Beginning with small, in time you can reach large peaks.