Mathematics is the science that builds the world. As a scientist, and a simple person - no one can do without it. First, young children are taught to count, then add, subtract, multiply, and divide, letters come into secondary school in high school, and in the older one, they cannot do without them.
But today we will talk about what all well-known mathematics is built on. On the community of numbers called "sequence limits".
What are sequences and where is their limit?
The meaning of the word "sequence" is not difficult to interpret. This is the construction of things where someone or something is arranged in a certain order or queue. For example, the queue for tickets to the zoo is a sequence. And it can be only one! If, for example, to look at the queue at the store - this is one sequence. And if one person suddenly leaves this line, 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 such values on a number line to which a sequence of numbers tends. Why seeks, and does not end? Everything is simple, the number line has no end, and most of the sequences, like rays, have only the beginning and look like this:
Hence the definition of a sequence is a function of the natural argument. In simpler words, it is a series of members of some set.
How is a numeric sequence built?
The simplest example of a numerical sequence might look like this: 1, 2, 3, 4, ... n ...
In most cases, for practical purposes, the sequences are built from numbers, and each next member of the series, denoted by X, has its own name. For example:
x1 - the first member of the sequence;
x2 - the second term of the sequence;
x3 - the third member;
xn - the nth member.
In practical methods, the sequence is given by the general formula, in which there is some variable. For example:
Xn = 3n, then the number series itself will look like this:
You should not forget that in the general record 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 plunge deeper into the very concept of such a numerical series, which everyone encountered while in the middle classes. An arithmetic progression is a series of numbers in which the difference between neighboring members is constant.
Task: “Let a1 = 15, and the step of the progression of the number series is d = 4. Build the first 4 members of this series. ”
Solution: a1 = 15 (by condition) - the first member of the progression (number series).
but2 = 15 + 4 = 19 - the second member of the progression.
but3 = 19 + 4 = 23 - the third member.
but4 = 23 + 4 = 27 - the fourth member.
Types of sequences
Most of the sequences are endless, it's worth remembering for a lifetime. There are two interesting types of number series. The first is given by the formula an = (- 1) n. Mathematicians often call this sequence a flasher. Why? Check its numerical series.
-1, 1, -1. 1, -1, 1, etc. With such an example, it becomes clear that numbers in sequences can be easily repeated.
Factorial sequence. It is easy to guess - factorial is present in the formula that defines the sequence. For example: an = (n + 1)!
Then the sequence will look like this:
but3 = 1x2x3x4 = 24, etc.
There is even a sequence consisting of the same number. So welln = 6 consists of an infinite set of sixes.
Definition of the sequence limit
The limits of sequences have long existed in mathematics. Of course, they deserved their own competent design. So, time to learn the definition of the limits of sequences. First, consider in detail the limit for a linear function:
- All limits are abbreviated lim.
- The limit entry consists of the abbreviation 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 certain number to which all members of the sequence infinitely approach. A simple example: ax = 4x + 1. Then the sequence itself will look like this.
5, 9, 13, 17, 21 ... x ...
Thus, this sequence will increase infinitely, and, therefore, its limit is equal to infinity as x → ∞, and this should be written as follows:
If we take a similar sequence, but x will tend to 1, we get:
A series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number more and more close to one (0.1, 0.2, 0.9, 0.986). From this series it can be seen that the limit of the 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 examined the limit of a numerical sequence, its definition and examples, we can proceed to a more complex topic. Absolutely all the limits of sequences can be formulated with one formula, which is usually analyzed in the first semester.
So, what does this set of letters, modules and inequalities signify?
∀ - the quantifier of universality, replacing the phrase "for all", "for all", etc.
∃ is a quantifier of existence, in this case means that there exists some value N belonging to the set of natural numbers.
A long vertical stick, following N, means that a given set of N is “such that.” In practice, it can mean "such that", "such that", etc.
Next comes the module. Obviously, a module is a distance which, by definition, cannot be negative. The difference modulus is strictly less than epsilon.
To consolidate the material, read the formula out loud.
Uncertainty and definiteness of the limit
The method for finding the limit of sequences, which was discussed above, is simple in use, but not so rational in practice. Try to find the limit for such a function:
If we substitute different values of "X" (each time increasing: 10, 100, 1000, etc.), then in the numerator we get ∞, but in the denominator also ∞. It turns out 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 power in the numerator of a fraction — it is 1, since x can be represented as x 1.
Now we find the highest power in the denominator. Also 1.
We divide both the numerator and the denominator into a variable in the highest degree. In this case, the fraction divided by x 1.
Next, we will find what value each addend containing the variable tends to. In this case, fractions are considered. As x → ∞, the value of each of the fractions tends to zero. When writing the work in writing, it is worth making the following footnotes:
The following expression is obtained:
Of course, fractions containing x did not become zeros! But their value is so small that it is completely allowed not to take it into account when calculating. In fact, x will never be equal to 0 in this case, because zero cannot be divided.
What is a neighborhood?
Suppose a professor has at his disposal a complex sequence, given, obviously, by a no less complex formula. The professor found the answer, but is he 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 the neighborhood.
Suppose that there exists a point a, its neighborhood in both directions on the number line is ε ("epsilon"). Since the last variable is distance, its value is always positive.
Now let's set some sequence xn and suppose that the tenth term in the sequence (x10 ) enters the neighborhood of a. How to write this fact in mathematical language?
Let's say x10 is located to the right of the point a, then the distance x10 –A \u0026 lt; ε, however, if you put the “X-tenth” to the left of point a, then the distance will be negative, but this is impossible, then you should add the left side of the inequality to the module. It turns out | x10 –And | \u0026 lt; ε.
Now it's time to clarify in practice that formula, which was mentioned above. A certain number a can be called the end point of a sequence if the inequality ε \u0026 gt; 0 holds for any limit of it, and the whole neighborhood has its natural number N such that all members of the sequence with more significant numbers will be inside the sequence | xn - a | \u0026 lt; ε.
With such knowledge, it is easy to implement the solution of the limits of a sequence, to prove or disprove a ready answer.
Theorems on the limits of sequences are an important component of the theory, without which practice is impossible. There are only four main theorems, remembering which, it is possible at times to facilitate the course of a solution or proof:
- The uniqueness of the limit of the 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 limited.
- The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
- The limit of the quotient from the division of two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.
Proof of sequences
Sometimes it is required to solve an inverse problem, to prove a given limit of a numerical sequence. Consider an example.
Prove that the limit of a sequence given by the formula is zero.
According to the rule discussed above, for any sequence the inequality | xn - a | \u0026 lt; ε. Substitute the specified value and point of reference. We get:
Express n through "epsilon" to show the existence of a certain number and prove the existence of a limit sequence.
At this stage, it is important to recall that "epsilon" and "en" are positive and non-zero numbers. Now it is possible to continue further transformations using the knowledge of inequalities obtained in high school.
With such a convenient method, one can prove the limit of a numerical sequence, however complex it may seem at first glance. The main thing - do not panic at the sight of the job.
Or maybe it is not?
The existence of a limit sequence is optional in practice. You can easily 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 numbers, cyclically repeating, cannot have a limit.
The same history is repeated with sequences consisting of a single number, fractional, having in the course of calculations the uncertainty of any order (0/0, ∞ / ∞, ∞ / 0, etc.). However, it should be remembered that the wrong calculation also takes place. Sometimes the limit of the sequences will help to recheck their own solutions.
Above we have considered several examples of sequences, methods for solving them, and now we will try to take a more specific case and call it “monotone sequence”.
Definition: it is fair to call any sequence monotonously increasing if the strict inequality x holds for it.n \u0026 lt; xn+1. Also, any sequence is rightly called monotone decreasing if for it the inequality xn \u0026 gt; xn+1.
Along with these two conditions, there are also similar weak inequalities. Accordingly, xn ≤ xn+1 (non-decreasing sequence) and xn ≥ xn+1 (non-increasing sequence).
But it is easier to understand this with examples.
The sequence given by xn = 2 + n, forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.
Limit of convergent and limited sequence
Bounded sequence - a sequence that has a limit. A converging sequence is a series of numbers that has an infinitely small 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 quantity (real or complex). If you draw a sequence diagram, then at a certain point it will seem to converge, to strive to turn to a certain value. Hence the name - convergent sequence.
The limit for such a sequence may or may not be. At first it is useful to understand when it is, from which one can push off when proving the absence of a limit.
Among the monotonous sequences emit converging and diverging. Converging is a sequence that is formed by the set x and has a real or complex limit in a given set. Divergent - a sequence that has no limit in its set (neither real nor complex).
Moreover, the sequence converges if its geometrical image converges its upper and lower limits.
The limit of a converging sequence in many cases can be equal to zero, since any infinitely small sequence has a known limit (zero).
No matter what converging sequence, they are all limited, but not all bounded sequences converge.
The sum, difference, product of two converging sequences is also a convergent sequence. However, the quotient can also be convergent if it is defined!
Various actions with outside
The limits of sequences are as substantial (in most cases) as the 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 is true: 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 the quotient of two sequences is equal to the quotient of their limits, provided that the limit is not equal to zero. After all, if the limit of sequences is equal to zero, then we get a division by zero, which is impossible.
- The sum of any quantity of arbitrarily small quantities will also be a small quantity.
- The sum of any number of large quantities will be an infinitely large quantity.
- The product of arbitrarily small quantities is infinitely small.
- The product of any number of large numbers is an infinitely large value.
- If the initial sequence tends to an infinitely large number, then the quantity opposite to it will be infinitely small and tend to zero.
In fact, calculating the limit of a sequence is not such a difficult task if you know a simple algorithm. But the limits of sequences is a topic that requires maximum attention and perseverance. Of course, it is enough just to catch the essence of the solution of such expressions. Starting from small, over time you can reach large peaks.