Math — science building peace. As a scientist and a simple man — no one can do without it. First, young children learn to count, then add, subtract, multiply and divide to the middle school to take the course letter designations, and the eldest without them is not enough.
But today we will talk about what the whole of known mathematics. About the community of numbers called "limits of sequences".
What is the sequence and where is their limit?
The meaning of the word "sequence" to interpret easily. It is building things where someone or something arranged in a particular order or queue. For example, the queue for tickets to the zoo is a sequence. And it can be only one! If, for example, look at the queue at the shop is one sequence. But if one person in the queue will leave suddenly, it's a different place, a different order.
The word "limit" is also easily interpreted is the end of something. However, in mathematics, limits of sequences — these are the values on the number line, which seeks a sequence of numbers. Why are aims, not ends? Quite simply, the number line has no end, and most of the sequences, like the rays are just the beginning and are as follows:
Hence the definition of the sequence — a function of natural argument. More simple words is the number of members of some set.
How is the number pattern?
A simple example of a numerical sequence might look like this: 1, 2, 3, 4, ...n...
In most cases, for practical purposes, sequences are constructed from numbers, and each following member of a number, denote it by X, has its name. For example:
x1 the first term of the sequence
x2 — the second term of the sequence
x3 the third member
xn is the nth member.
In practice the sequence is a common formula, which includes some variable. For example:
Xn =3n, then the series of numbers would look like this:
It should not be forgotten that in the total sequence records, you can use any letters, not just X. for Example: y, z, k, etc.
An arithmetic progression as part of a sequence
Before looking for the limits of sequences, it is advisable to dip a little deeper into the notion of such a numerical range, with which all are faced, being in middle school. An arithmetic progression is a series of numbers in which the difference between adjacent members is constant.
Problem: "Let a1 =15, and the step of progression of the numerical range of d=4. Build the first 4 members of this series"
The solution: a1 = 15 (the condition) is the first member of the progression (numerical series).
and2 = 15 4=19 is the second member of the progression.
and3 =19 4=23 is the third member.
and4 =23 4=27 — the fourth member.
The types of sequences
Most of the sequence is infinite, it is worth remembering for a lifetime. There are two interesting species of the numerical range. The first is given by the formula andn =(-1) n. Mathematicians often call this sequence flasher. Why? Check its numerical range.
-1, 1, -1. 1, -1, 1 etc. In such example, it becomes clear that the numbers in the sequence can easily be repeated.
Factorial sequence. It is easy to guess — in the formula defining the sequence is present in the factorial. For example: n = (n 1)!
Then the sequence will look like the following:
and3 = 1х2х3х4 =24, etc.
There is even a sequence consisting of the same number. So, andn =6 consists of an infinite number of sixes.
Definition of limit of a sequence
Limits of sequences have long existed in mathematics. Of course, they deserve their own proper design. So, time to learn the definition of limits of sequences. To begin, consider in detail the limit for the linear function:
- All limits are indicated by the abbreviation lim.
- The entry limit is from the abbreviation lim, a variable, tending to a certain number, zero or infinity, and also of the function itself.
It is easy to understand that the definition of limit of a sequence can be formulated as follows: is some number to which it is infinitely approaching all members of the sequence. A simple example: x = 4x 1. Then the sequence itself will look like the following.
5, 9, 13, 17, 21...x ...
Thus, this sequence will endlessly increase and, hence, its limit is equal to infinity at x→∞, and write it as follows:
If we take a similar series, but x will tend to 1, we get:
And series of numbers will be: 1.4, 1.8, 4.6, 4.944, etc. Every time you need to substitute a number more close to the unit (0.1, 0.2, 0.9, 0.986). This series shows that the limit of the function is five.
From this part it is necessary to remember, what is the limit of numerical sequence, the definition and method of solving simple tasks.
The General designation of the limit of sequences
Having examined the limit of numerical sequence, the definition of it and examples, you can proceed to more difficult subject. All the limits of sequences can be formulated as a single equation, which is usually dismantled in the first semester.
So, what does this collection of letters, modules and signs of the inequalities?
∀ — universal quantifier replacing the phrase "for all", "all", etc.
∃ — quantifier of existence, in this case indicates that there is some value of N belonging to the set of natural numbers.
Long vertical stick, the next over N, so that the set N "such that". In practice, it can mean "such that", "such that", etc.
Next is the module. Obviously, the module is the distance, which by definition cannot be negative. So the absolute difference is less than “Epsilon”.
To consolidate the material, read the formula aloud.
Uncertainty and certainty limit
Method of finding the limit of the sequences discussed above, albeit simple to use, but not so rational in practice. Try to find the limit for here such function:
If we substitute different values of "x" (each time increasing: 10, 100, 1000, etc.), then the numerator will receive ∞, but the denominator is also ∞. It is quite strange roll: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 older degree of the numerator of the fraction is 1, since x can be represented as x 1 .
Now we find the oldest degree in the denominator. Also 1.
Divide the numerator and the denominator by the variable in the highest degree. In this case, a fraction is divided by x 1 .
Next, find which of the values committed each term containing the variable. In this case, fractions are considered. As x→∞ the value of each of the fractions tends to zero. When making the work written should do the following footnote:
We get the following expression:
Of course, the fraction that contains the x, were not zeros! But their value is so small that it is not allowed to take it into account in the calculations. In fact, x will never equal 0 in this case, because zero cannot be split.
What is a neighborhood?
Suppose a Professor complex sequence, set, obviously, is no less complex formula. The Professor found the answer, but is it right? After all, people make mistakes.
Auguste Cauchy in his time came up with a great way to prove the limits of sequences. His method is called the operating surroundings.
Suppose that there is some point and its neighborhood in both directions on the number line is equal to ε ("Epsilon"). Because the last variable is distance, its value is always positive.
Now ask some sequence xn and put that the tenth term of the sequence (x10 ) is included in the neighborhood. How to record this fact in mathematical language?
Assume x10 located to the right of point a, then the distance x10 and10 –|
Now it's time to clarify in practice the formula, as discussed above. A certain number and fair to describe the end point of the sequence if for any its limit the following inequality holds εn – a|
With such knowledge easily implement the solution within a sequence, to prove or disprove a ready answer.
Theorems about limits of sequences is an important component of the theory, which is essential practice. There are only four main theorem, remembering that can significantly facilitate the progress of the solution or proof:
- Uniqueness of limit of a sequence. The limit of any sequence can be only one, or none. The same example with the queue, which can be only one end.
- If a series of numbers has a limit, then the sequence of these numbers is limited.
- The limit of the sum (difference, product) of the sequence is equal to the sum (difference, product) of their limits.
- The limit of the private from division of two sequences is equal to the quotient of the limits, and then only when the denominator becomes zero.
Proof of sequences
Sometimes it is necessary to solve the inverse problem, prove the limit of a numerical sequence. Consider the following example.
To prove that the limit of a sequence defined by the formula equal to zero.
According to the above rule, any sequence should be performed inequality |xn – a|
Express n via Epsilon to show the existence of some numbers and prove the existence of limit of a sequence.
At this stage it is important to recall that the "Epsilon" and "n" – numbers are positive and not equal to zero. You can now continue further transformation, using knowledge of inequalities obtained in high school.
This is such a convenient method to prove the limit of numerical sequence, as difficult as it at first glance. The main thing — not to panic at the sight of the task.
Maybe it is not?
The existence of the limit of a sequence, not necessarily in practice. It is easy to find such a series of numbers that really have no end. For example, the same "flasher" xn = (-1) n. it is obvious that the sequence consisting of only two digits, cyclically repeated, cannot have limit.
The same story is repeated with sequences consisting of a single number, a fraction, having in the course of computing the uncertainty of any order ( 0/0, ∞/∞, ∞/0 etc.). However, remember that a wrong calculation is also the place to be. Sometimes the limit posledovatel find will help double-check their own solutions.
Above we have considered several examples of sequences, methods of their solution, and now try to take a more specific case and call it "monotone sequence".
Definition: any sequence rightly be called monotonically increasing if it meets the strict inequality xn n+1. Also any sequence rightly be called monotone decreasing if it meets the inequality xn n+1.
Along with these two conditions also exist such non-strict inequality. Accordingly, xn ≤ xn+1 (a non-decreasing sequence) and xn ≥ xn+1 (nonincreasing sequence).
But it's easier to understand like in the examples.
The sequence defined by formula xn = 2 n, forming the following series of numbers: 4, 5, 6, etc. It is a monotonically increasing sequence.
The limit of a convergent and a bounded sequence
A bounded sequence is a sequence having a limit. Convergent sequence — a series of numbers having infinitely small in the limit.
Thus, the limit of a bounded sequence is any real or complex number. Remember that the limit can be only one.
The limit of a convergent sequence is the magnitude of the infinitesimal (real or complex). If you draw a sequence diagram, then at a certain point it will be like to come together, to seek to apply to a certain value. Hence the name — converging sequence.
The limit of a monotonic sequence
The limit of such a sequence may or may not be. It is useful first to understand when it is, it is possible to build the proof of the absence limit.
Among the monotone sequences distinguish convergent and divergent. Convergent is such a sequence, which is composed of x and has a given real or complex limit. A divergent sequence has no limit in its variety (neither valid nor comprehensive).
Moreover, the sequence converges if the geometric image of the upper and lower limits converge.
The limit of a convergent sequence in many cases may be zero, since any infinitesimal sequence has a known limit (zero).
What a convergent sequence take, they are all limited, however, not all bounded sequences converge.
The sum, difference, product of two convergent sequences is also convergent sequence. However, the private can also be convergent, if it is defined!
Various actions with outside
Limits of sequences is a significant (in most cases) the magnitude and numbers and numbers: 1, 2, 15, 24, 362 etc. it Turns out that the limits you can carry out some operations.
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, on the basis of the fourth theorem about limits of sequences, fairly following equality: limit of the product n-th number of the sequence is equal to the product of their limits. The same is true for division: the limit of a private of two sequences equal to the quotient of their limits, provided the limit is not zero. After all, if the limit of the sequence is equal to zero, we get divide by zero, which is impossible.
The properties of the values of the sequence
- The sum of any number of indefinitely small quantities is also of small value.
- The amount of any number large quantities will be infinitely large value.
- The product of arbitrarily small quantities infinitely small.
- Work as many large numbers — the value infinitely large.
- If the original sequence tends to an infinitely large number, then it reverse value will be infinitesimally small and tends to zero.
Actually calculate the limit of the 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, just enough to capture the essence of solving such expressions. Starting small, over time you can achieve great heights.