How to create a number sequence formula. Number sequence. How to find the limit of a sequence

Vida y= f(x), x ABOUT N, Where N– a set of natural numbers (or a function of a natural argument), denoted y=f(n) or y 1 ,y 2 ,…, y n,…. Values y 1 ,y 2 ,y 3 ,… are called respectively the first, second, third, ... members of the sequence.

For example, for the function y= n 2 can be written:

y 1 = 1 2 = 1;

y 2 = 2 2 = 4;

y 3 = 3 2 = 9;…y n = n 2 ;…

Methods for specifying sequences. Sequences can be specified different ways, among which three are especially important: analytical, descriptive and recurrent.

1. A sequence is given analytically if its formula is given n th member:

y n=f(n).

Example. y n= 2n – 1 – sequence of odd numbers: 1, 3, 5, 7, 9, …

2. Descriptive way of setting number sequence is that it explains from which elements the sequence is built.

Example 1. “All terms of the sequence are equal to 1.” This means, we're talking about about the stationary sequence 1, 1, 1, …, 1, ….

Example 2. “A sequence consists of all prime numbers in ascending order". Thus, the given sequence is 2, 3, 5, 7, 11, …. With this method of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to.

3. The recurrent method of specifying a sequence is to specify a rule that allows you to calculate n-th member of a sequence if its previous members are known. The name recurrent method comes from the Latin word recurrent- come back. Most often, in such cases, a formula is indicated that allows one to express n th member of the sequence through the previous ones, and specify 1–2 initial members of the sequence.

Example 1. y 1 = 3; y n = y n–1 + 4 if n = 2, 3, 4,….

Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; ….

You can see that the sequence obtained in this example can also be specified analytically: y n= 4n – 1.

Example 2. y 1 = 1; y 2 = 1; y n = y n –2 + y n–1 if n = 3, 4,….

Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8;

The sequence in this example is especially studied in mathematics because it has a number of interesting properties and applications. It is called the Fibonacci sequence, named after the 13th century Italian mathematician. It is very easy to define the Fibonacci sequence recurrently, but very difficult analytically. n The th Fibonacci number is expressed through its serial number by the following formula.

At first glance, the formula for n the th Fibonacci number seems implausible, since the formula defining the sequence of natural numbers only contains square roots, but you can check “manually” the validity of this formula for the first few n.

Properties of number sequences.

A numerical sequence is a special case of a numerical function, therefore a number of properties of functions are also considered for sequences.

Definition . Subsequence ( y n} is called increasing if each of its terms (except the first) is greater than the previous one:

y 1 y 2 y 3 y n y n +1

Definition.Sequence ( y n} is called decreasing if each of its terms (except the first) is less than the previous one:

y 1 > y 2 > y 3 > … > y n> y n +1 > … .

Increasing and decreasing sequences are combined under the common term - monotonic sequences.

Example 1. y 1 = 1; y n= n 2 – increasing sequence.

Thus, the following theorem is true (a characteristic property of an arithmetic progression). A number sequence is arithmetic if and only if each of its members, except the first (and the last in the case of a finite sequence), is equal to the arithmetic mean of the preceding and subsequent members.

Example. At what value x numbers 3 x + 2, 5x– 4 and 11 x+ 12 form a finite arithmetic progression?

According to the characteristic property, the given expressions must satisfy the relation

5x – 4 = ((3x + 2) + (11x + 12))/2.

Solving this equation gives x= –5,5. At this value x given expressions 3 x + 2, 5x– 4 and 11 x+ 12 take, respectively, the values –14.5, –31,5, –48,5. This is an arithmetic progression, its difference is –17.

Geometric progression.

A numerical sequence, all terms of which are non-zero and each term of which, starting from the second, is obtained from the previous term by multiplying by the same number q, is called a geometric progression, and the number q- the denominator of a geometric progression.

Thus, a geometric progression is a number sequence ( b n), defined recursively by the relations

b 1 = b, b n = b n –1 q (n = 2, 3, 4…).

(b And q – given numbers, b ≠ 0, q ≠ 0).

Example 1. 2, 6, 18, 54, ... – increasing geometric progression b = 2, q = 3.

Example 2. 2, –2, 2, –2, … – geometric progression b= 2,q= –1.

Example 3. 8, 8, 8, 8, … – geometric progression b= 8, q= 1.

A geometric progression is an increasing sequence if b 1 > 0, q> 1, and decreasing if b 1 > 0, 0 q

One of the obvious properties of a geometric progression is that if the sequence is a geometric progression, then so is the sequence of squares, i.e.

b 1 2 , b 2 2 , b 3 2 , …, b n 2,... is a geometric progression whose first term is equal to b 1 2 , and the denominator is q 2 .

Formula n- the th term of the geometric progression has the form

b n= b 1 qn– 1 .

You can obtain a formula for the sum of terms of a finite geometric progression.

Let a finite geometric progression be given

b 1 ,b 2 ,b 3 , …, b n

let S n – the sum of its members, i.e.

S n= b 1 + b 2 + b 3 + … +b n.

It is accepted that q No. 1. To determine S n an artificial technique is used: some geometric transformations of the expression are performed S n q.

S n q = (b 1 + b 2 + b 3 + … + b n –1 + b n)q = b 2 + b 3 + b 4 + …+ b n+ b n q = S n+ b n q– b 1 .

Thus, S n q= S n +b n q – b 1 and therefore

This is the formula with umma n terms of geometric progression for the case when q≠ 1.

At q= 1 the formula need not be derived separately; it is obvious that in this case S n= a 1 n.

The progression is called geometric because each term in it, except the first, is equal to the geometric mean of the previous and subsequent terms. Indeed, since

bn=bn- 1 q;

bn = bn+ 1 /q,

hence, b n 2=bn– 1 bn+ 1 and the following theorem is true (a characteristic property of a geometric progression):

a number sequence is a geometric progression if and only if the square of each of its terms, except the first (and the last in the case of a finite sequence), equal to the product previous and subsequent members.

Consistency limit.

Let there be a sequence ( c n} = {1/n}. This sequence is called harmonic, since each of its terms, starting from the second, is the harmonic mean between the previous and subsequent terms. Geometric mean of numbers a And b there is a number

Otherwise the sequence is called divergent.

Based on this definition, one can, for example, prove the existence of a limit A=0 for the harmonic sequence ( c n} = {1/n). Let ε be arbitrarily small positive number. The difference is considered

Does such a thing exist? N that's for everyone n ≥ N inequality 1 holds /N ? If we take it as N any natural number, exceeding 1/ε , then for everyone n ≥ N inequality 1 holds /n ≤ 1/N ε , Q.E.D.

Proving the presence of a limit for a particular sequence can sometimes be very difficult. The most frequently occurring sequences are well studied and are listed in reference books. There are important theorems that allow you to conclude that a given sequence has a limit (and even calculate it), based on already studied sequences.

Theorem 1. If a sequence has a limit, then it is bounded.

Theorem 2. If a sequence is monotonic and bounded, then it has a limit.

Theorem 3. If the sequence ( a n} has a limit A, then the sequences ( ca n}, {a n+ c) and (| a n|} have limits cA, A +c, |A| accordingly (here c– arbitrary number).

Theorem 4. If the sequences ( a n} And ( b n) have limits equal to A And B pa n + qbn) has a limit pA+ qB.

Theorem 5. If the sequences ( a n) And ( b n)have limits equal to A And B accordingly, then the sequence ( a n b n) has a limit AB.

Theorem 6. If the sequences ( a n} And ( b n) have limits equal to A And B accordingly, and, in addition, b n ≠ 0 and B≠ 0, then the sequence ( a n / b n) has a limit A/B.

Anna Chugainova

For example, for the function y= n 2 can be written:

y 1 = 1 2 = 1;

y 2 = 2 2 = 4;

y 3 = 3 2 = 9;…y n = n 2 ;…

Methods for specifying sequences. Sequences can be specified in various ways, among which three are especially important: analytical, descriptive and recurrent.

1. A sequence is given analytically if its formula is given n th member:

y n=f(n).

Example. y n= 2n – 1 – sequence of odd numbers: 1, 3, 5, 7, 9, …

2. Descriptive The way to specify a numerical sequence is to explain from which elements the sequence is built.

Example 1. “All terms of the sequence are equal to 1.” This means we are talking about a stationary sequence 1, 1, 1, …, 1, ….

Example 2: “The sequence consists of all prime numbers in ascending order.” Thus, the given sequence is 2, 3, 5, 7, 11, …. With this method of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to.

Example 1. y 1 = 3; y n = y n–1 + 4 if n = 2, 3, 4,….

Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; ….

You can see that the sequence obtained in this example can also be specified analytically: y n= 4n – 1.

Example 2. y 1 = 1; y 2 = 1; y n = y n –2 + y n–1 if n = 3, 4,….

Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8;

Properties of number sequences.

A numerical sequence is a special case of a numerical function, therefore a number of properties of functions are also considered for sequences.

Definition . Subsequence ( y n} is called increasing if each of its terms (except the first) is greater than the previous one:

y 1 y 2 y 3 y n y n +1

Definition.Sequence ( y n} is called decreasing if each of its terms (except the first) is less than the previous one:

y 1 > y 2 > y 3 > … > y n> y n +1 > … .

Increasing and decreasing sequences are combined under the common term - monotonic sequences.

Example 1. y 1 = 1; y n= n 2 – increasing sequence.

Example. At what value x numbers 3 x + 2, 5x– 4 and 11 x+ 12 form a finite arithmetic progression?

According to the characteristic property, the given expressions must satisfy the relation

5x – 4 = ((3x + 2) + (11x + 12))/2.

Geometric progression.

Thus, a geometric progression is a number sequence ( b n), defined recursively by the relations

b 1 = b, b n = b n –1 q (n = 2, 3, 4…).

(b And q – given numbers, b ≠ 0, q ≠ 0).

Example 1. 2, 6, 18, 54, ... – increasing geometric progression b = 2, q = 3.

Example 2. 2, –2, 2, –2, … – geometric progression b= 2,q= –1.

Example 3. 8, 8, 8, 8, … – geometric progression b= 8, q= 1.

A geometric progression is an increasing sequence if b 1 > 0, q> 1, and decreasing if b 1 > 0, 0 q

One of the obvious properties of a geometric progression is that if the sequence is a geometric progression, then so is the sequence of squares, i.e.

b 1 2 , b 2 2 , b 3 2 , …, b n 2,... is a geometric progression whose first term is equal to b 1 2 , and the denominator is q 2 .

Formula n- the th term of the geometric progression has the form

b n= b 1 qn– 1 .

You can obtain a formula for the sum of terms of a finite geometric progression.

Let a finite geometric progression be given

b 1 ,b 2 ,b 3 , …, b n

let S n – the sum of its members, i.e.

S n= b 1 + b 2 + b 3 + … +b n.

It is accepted that q No. 1. To determine S n an artificial technique is used: some geometric transformations of the expression are performed S n q.

S n q = (b 1 + b 2 + b 3 + … + b n –1 + b n)q = b 2 + b 3 + b 4 + …+ b n+ b n q = S n+ b n q– b 1 .

Thus, S n q= S n +b n q – b 1 and therefore

This is the formula with umma n terms of geometric progression for the case when q≠ 1.

At q= 1 the formula need not be derived separately; it is obvious that in this case S n= a 1 n.

The progression is called geometric because each term in it, except the first, is equal to the geometric mean of the previous and subsequent terms. Indeed, since

bn=bn- 1 q;

bn = bn+ 1 /q,

hence, b n 2=bn– 1 bn+ 1 and the following theorem is true (a characteristic property of a geometric progression):

a number sequence is a geometric progression if and only if the square of each of its terms, except the first (and the last in the case of a finite sequence), is equal to the product of the previous and subsequent terms.

Consistency limit.

Otherwise the sequence is called divergent.

Based on this definition, one can, for example, prove the existence of a limit A=0 for the harmonic sequence ( c n} = {1/n). Let ε be an arbitrarily small positive number. The difference is considered

Does such a thing exist? N that's for everyone n ≥ N inequality 1 holds /N ? If we take it as N any natural number greater than 1/ε , then for everyone n ≥ N inequality 1 holds /n ≤ 1/N ε , Q.E.D.

Theorem 1. If a sequence has a limit, then it is bounded.

Theorem 2. If a sequence is monotonic and bounded, then it has a limit.

Theorem 3. If the sequence ( a n} has a limit A, then the sequences ( ca n}, {a n+ c) and (| a n|} have limits cA, A +c, |A| accordingly (here c– arbitrary number).

Theorem 4. If the sequences ( a n} And ( b n) have limits equal to A And B pa n + qbn) has a limit pA+ qB.

Theorem 5. If the sequences ( a n) And ( b n)have limits equal to A And B accordingly, then the sequence ( a n b n) has a limit AB.

Theorem 6. If the sequences ( a n} And ( b n) have limits equal to A And B accordingly, and, in addition, b n ≠ 0 and B≠ 0, then the sequence ( a n / b n) has a limit A/B.

Anna Chugainova

Mathematics is the science that builds the world. Both the scientist and the common man - no one can do without it. First, young children are taught to count, then to add, subtract, multiply and divide, to high school come into play letter designations, and in the older age you can’t do without them.

But today we will talk about what all known mathematics is based on. About a 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 an arrangement of things where someone or something is located in a certain order or queue. For example, the queue for tickets to the zoo is a sequence. And there can only be one! If, for example, you look at the queue at the store, this is one sequence. And if one person from this queue suddenly leaves, then this is a different queue, a different order.

The word “limit” is also easily interpreted - it is the end of something. However, in mathematics, the limits of sequences are those values on the number line to which a sequence of numbers tends. Why does it strive and not end? It's simple, the number line has no end, and most sequences, like rays, have only a beginning and look like this:

x 1, x 2, x 3,...x n...

Hence the definition of a sequence is a function of the natural argument. More in simple words is a series of members of a certain set.

How is the number sequence constructed?

A simple example of a number sequence might look like this: 1, 2, 3, 4, …n…

In most cases, for practical purposes, sequences are built from numbers, and each next member of the series, let's denote it by X, has its own name. For example:

x 1 is the first member of the sequence;

x 2 is the second term of the sequence;

x 3 is the third term;

x n is the nth term.

In practical methods, the sequence is given by a general formula in which there is a certain variable. For example:

X n =3n, then the series of numbers itself will look like this:

It is worth remembering that when writing sequences in general, 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 number series, which everyone encountered when they were in middle school. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.

Problem: “Let a 1 = 15, and the progression step of the number series d = 4. Construct the first 4 terms of this series"

Solution: a 1 = 15 (by condition) is the first term of the progression (number series).

and 2 = 15+4=19 is the second term of the progression.

and 3 =19+4=23 is the third term.

and 4 =23+4=27 is the fourth term.

However similar method it is difficult to reach large values, for example up to a 125. . Especially for such cases, a formula convenient for practice was derived: a n =a 1 +d(n-1). In this case, a 125 =15+4(125-1)=511.

Types of sequences

Most of the sequences are endless, it's worth remembering for the rest of your life. There are two interesting looking number series. The first is given by the formula a n =(-1) n. Mathematicians often call this sequence a flasher. Why? Let's check its number series.

1, 1, -1, 1, -1, 1, etc. With an example like this, it becomes clear that numbers in sequences can easily be repeated.

Factorial sequence. It's easy to guess - the formula defining the sequence contains a factorial. For example: a n = (n+1)!

Then the sequence will look like this:

a 2 = 1x2x3 = 6;

and 3 = 1x2x3x4 = 24, etc.

Sequence given arithmetic progression, is called infinitely decreasing if the inequality -1 is observed for all its terms

and 3 = - 1/8, etc.

There is even a sequence consisting of the same number. So, n =6 consists of an infinite number of sixes.

Determining the Sequence Limit

Sequence limits have long existed in mathematics. Of course, they deserve their own competent design. So, time to learn the definition of sequence limits. First, let's look at the limit for a linear function in detail:

All limits are abbreviated as lim.
The notation for a limit consists of the abbreviation lim, any variable tending to a certain number, zero or infinity, as well as 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: a x = 4x+1. Then the sequence itself will look like this.

5, 9, 13, 17, 21…x…

Thus, this sequence will increase indefinitely, which means its limit is equal to infinity as x→∞, and it should be written like this:

If we take a similar sequence, but x tends to 1, we get:

And the series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number closer to one (0.1, 0.2, 0.9, 0.986). From this series it is clear that the limit of the function is five.

From this part it is worth remembering what the limit of a numerical sequence is, the definition and method for solving simple problems.

General designation for the limit of sequences

Having examined the limit of a number sequence, its definition and examples, you can proceed to a more complex topic. Absolutely all limits of sequences can be formulated by one formula, which is usually analyzed in the first semester.

So, what does this set of letters, modules and inequality signs mean?

∀ is a universal quantifier, replacing the phrases “for all”, “for everything”, etc.

∃ is an existential quantifier, in this case it means that there is some value N belonging 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”, etc.

To reinforce the material, read the formula out loud.

Uncertainty and certainty of the limit

The method of finding the limit of sequences, which was discussed above, although simple to use, is not so rational in practice. Try to find the limit for this function:

If we substitute different values of “x” (increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. This results in a rather strange fraction:

But is this really so? Calculating the limit of a number sequence in this case seems quite easy. It would be possible to leave everything as it is, because the answer is ready, and it was received under reasonable conditions, but there is another way specifically for such cases.

First, let's find the highest degree in the numerator of the fraction - this is 1, since x can be represented as x 1.

Now let's find the highest degree in the denominator. Also 1.

Let's divide both the numerator and the denominator by the variable to the highest degree. In this case, divide the fraction by x 1.

Next, we will find what value each term containing a variable tends to. In this case, fractions are considered. As x→∞, the value of each fraction tends to zero. When submitting your work in writing, you should make the following footnotes:

This results in the following expression:

Of course, the fractions containing x did not become zeros! But their value is so small that it is completely permissible not to take it into account in calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero.

What is a neighborhood?

Suppose the professor has at his disposal a complex sequence, given, obviously, by an equally complex formula. The professor has found the answer, but is it right? After all, all people make mistakes.

Auguste Cauchy once came up with an excellent way to prove the limits of sequences. His method was called neighborhood manipulation.

Suppose that there is a certain 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 let's define some sequence x n and assume that the tenth term of the sequence (x 10) is in the neighborhood of a. How can we write this fact in mathematical language?

Let's say x 10 is to the right of point a, then the distance x 10 -a<ε, однако, если расположить «икс десятое» левее точки а, то расстояние получится отрицательным, а это невозможно, значит, следует занести левую часть неравенства под модуль. Получится |х 10 -а|<ε.

Now it’s time to explain in practice the formula discussed above. It is fair to call a certain number a the end point of a sequence if for any of its limits the inequality ε>0 is satisfied, and the entire neighborhood has its own natural number N, such that all members of the sequence with higher numbers will be inside the sequence |x n - a|< ε.

With such knowledge it is easy to solve the sequence limits, prove or disprove the ready-made answer.

Theorems

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 can make the process of solving or proving much easier:

Uniqueness of the limit of a sequence. Any sequence can have only one limit or none at all. The same example with a queue that can only have 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 sequences is equal to the sum (difference, product) of their limits.
The limit of the quotient of dividing two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.

Proof of sequences

Sometimes you need to solve an inverse problem, to prove a given limit of a numerical sequence. Let's look at an example.

Prove that the limit of the sequence given by the formula is zero.

According to the rule discussed above, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим:

Let us express n through “epsilon” to show the existence of a certain number and prove the presence of a limit of the sequence.

At this point, it is important to remember that “epsilon” and “en” are positive numbers and are not equal to zero. Now it is possible to continue further transformations using the knowledge about inequalities gained in high school.

How does it turn out that n > -3 + 1/ε. Since it is worth remembering that we are talking about natural numbers, the result can be rounded by putting it in square brackets. Thus, it was proven that for any value of the “epsilon” neighborhood of the point a = 0, a value was found such that the initial inequality is satisfied. From here we can safely say that the number a is the limit of a given sequence. Q.E.D.

This convenient method can be used to prove the limit of a numerical sequence, no matter how complex it may be at first glance. The main thing is not to panic when you see the task.

Or maybe he's not there?

The existence of a consistency limit is not necessary in practice. You can easily come across series of numbers that really have no end. For example, the same “flashing light” x n = (-1) n. it is obvious that a sequence consisting of only two digits, repeated cyclically, cannot have a limit.

The same story is repeated with sequences consisting of one number, fractional ones, having uncertainty of any order during calculations (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculations also occur. Sometimes double-checking your own solution will help you find the sequence limit.

Monotonic sequence

Several examples of sequences and methods for solving them were discussed above, and now let’s try to take a more specific case and call it a “monotonic sequence.”

Definition: any sequence can rightly be called monotonically increasing if the strict inequality x n holds for it< x n +1. Также любую последовательность справедливо называть монотонной убывающей, если для неё выполняется неравенство x n >x n +1.

Along with these two conditions, there are also similar non-strict inequalities. Accordingly, x n ≤ x n +1 (non-decreasing sequence) and x n ≥ x n +1 (non-increasing sequence).

But it’s easier to understand this with examples.

The sequence given by the formula x n = 2+n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.

And if we take x n =1/n, we get the series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing sequence.

Limit of a convergent and bounded sequence

A bounded sequence is a sequence that has a limit. A convergent sequence is a series of numbers that has an infinitesimal limit.

Thus, the limit of a bounded sequence is any real or complex number. Remember that there can only be one limit.

The limit of a convergent sequence is an infinitesimal (real or complex) quantity. If you draw a sequence diagram, then at a certain point it will seem to converge, tend to turn into a certain value. Hence the name - convergent sequence.

Limit of a monotonic sequence

There may or may not be a limit to such a sequence. First, it is useful to understand when it exists; from here you can start when proving the absence of a limit.

Among monotonic sequences, convergent and divergent are distinguished. Convergent is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent is a sequence that has no limit in its set (neither real nor complex).

Moreover, the sequence converges if, in a geometric representation, its upper and lower limits converge.

The limit of a convergent sequence can be zero in many cases, since any infinitesimal sequence has a known limit (zero).

Whatever convergent sequence you take, they are all bounded, but not all bounded sequences converge.

The sum, difference, 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 as significant (in most cases) as digits and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits.

First, like digits and numbers, the limits of any sequence can be added and subtracted. Based on the third theorem on the limits of sequences, the following equality holds: the limit of the sum of sequences is 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 nth 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 zero. After all, if the limit of sequences is equal to zero, then division by zero will result, which is impossible.

Properties of sequence quantities

It would seem that the limit of the numerical sequence has already been discussed in some detail, but phrases such as “infinitely small” and “infinitely large” numbers are mentioned more than once. Obviously, if there is a sequence 1/x, where x→∞, then such a fraction is infinitesimal, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such quantities have their own characteristics. The properties of the limit of a sequence having any small or large values are as follows:

The sum of any number of any number of 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 infinitesimal.
The product of any number of large numbers is infinitely large.
If the original sequence tends to an infinitely large number, then its inverse will be infinitesimal 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 consistency are a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution to such expressions. Starting small, you can achieve great heights over time.

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Life sequence

SEQUENCE - numbers or elements arranged in an organized order. Sequences can be finite (having a limited number of elements) or infinite, such as the complete sequence of natural numbers 1, 2, 3, 4….… …

Scientific and technical encyclopedic dictionary

Definition:Numerical sequence is called a numeric given on the set N of natural numbers. For numerical sequences, usually instead of f(n) write a n and denote the sequence as follows: ( a n ). Numbers a 1 , a 2 , …, a n,… called elements of the sequence.

Usually the number sequence is determined by the task n th element or a recurrent formula by which each subsequent element is determined through the previous one. A descriptive way of specifying a numerical sequence is also possible. For example:

All members of the sequence are equal to "1". This means we are talking about a stationary sequence 1, 1, 1, …, 1, ….

The sequence consists of all prime numbers in ascending order. Thus, the given sequence is 2, 3, 5, 7, 11, …. With this method of specifying the sequence in this example, it is difficult to answer what, say, the 1000th element of the sequence is equal to.

With the recurrent method, indicate a formula that allows you to express n th member of the sequence through the previous ones, and specify 1–2 initial members of the sequence.

y 1 = 3; y n =y n-1 + 4 , If n = 2, 3, 4,…

Here y 1 = 3; y 2 = 3 + 4 = 7;y 3 = 7 + 4 = 11; ….

y 1 = 1; y 2 = 1; y n =y n-2 + y n-1 , If n = 3, 4,…

Here: y 1 = 1; y 2 = 1; y 3 = 1 + 1 = 2; y 4 = 1 + 2 = 3; y 5 = 2 + 3 = 5; y 6 = 3 + 5 = 8;

Sequence expressed by recurrence formula y n =y n-1 + 4 can also be specified analytically: y n= y 1 +4*(n-1)

Let's check: y2=3+4*(2-1)=7, y3=3+4*(3-1)=11

Here we do not need to know the previous member of the numerical sequence to calculate the nth element; we just need to specify its number and the value of the first element.

As we can see, this method of specifying a numerical sequence is very similar to the analytical method of specifying functions. In fact, a number sequence is a special type of number function, so a number of properties of functions can be considered for sequences as well.

Number sequences are a very interesting and educational topic. This topic is found in tasks of increased complexity that are offered to students by the authors of didactic materials, in problems of mathematical Olympiads, entrance exams to Higher Educational Institutions and. And if you want to learn more about the different types of number sequences, click here. Well, if everything is clear and simple to you, then try to answer.

Consider a series of natural numbers: 1, 2, 3, , n – 1, n,  .

If we replace every natural number n in this series by a certain number a n, following some law, we get a new series of numbers:

a 1 , a 2 , a 3 , , a n –1 , a n , ,

briefly designated and called numerical sequence. Magnitude a n is called a common member of a number sequence. Usually the number sequence is given by some formula a n = f(n) allowing you to find any member of the sequence by its number n; this formula is called the general term formula. Note that it is not always possible to define a numerical sequence using a general term formula; sometimes a sequence is specified by describing its members.

By definition, a sequence always contains an infinite number of elements: any two different elements differ at least in their numbers, of which there are infinitely many.

A number sequence is a special case of a function. A sequence is a function defined on the set of natural numbers and taking values in the set of real numbers, i.e. a function of the form f : N  R.

Subsequence
called increasing(decreasing), if for any n  N
Such sequences are called strictly monotonous.

Sometimes it is convenient to use not all natural numbers as numbers, but only some of them (for example, natural numbers starting from some natural number n 0). For numbering it is also possible to use not only natural numbers, but also other numbers, for example, n= 0, 1, 2,  (here zero is added as another number to the set of natural numbers). In such cases, when specifying the sequence, indicate what values the numbers take n.

If in some sequence for any n  N
then the sequence is called non-decreasing(non-increasing). Such sequences are called monotonous.

Example 1 . The number sequence 1, 2, 3, 4, 5, ... is a series of natural numbers and has a common term a n = n.

Example 2 . The number sequence 2, 4, 6, 8, 10, ... is a series of even numbers and has a common term a n = 2n.

Example 3 . 1.4, 1.41, 1.414, 1.4142, … – a numerical sequence of approximate values with increasing accuracy.

In the last example it is impossible to give a formula for the general term of the sequence.

Example 4 . Write the first 5 terms of a number sequence using its common term
. To calculate a 1 is needed in the formula for the general term a n instead of n substitute 1 to calculate a 2 − 2, etc. Then we have:

Test 6 . The common member of the sequence 1, 2, 6, 24, 120,  is:

Test 7 .
is:

Test 8 . Common member of the sequence
is:

Number sequence limit

Consider a number sequence whose common term approaches some number A when the serial number increases n. In this case, the number sequence is said to have a limit. This concept has a more strict definition.

Number A called the limit of a number sequence
:

(1)

if for any  > 0 there is such a number n 0 = n 0 (), depending on , which
at n > n 0 .

This definition means that A there is a limit to a number sequence if its common term approaches without limit A with increasing n. Geometrically, this means that for any  > 0 one can find such a number n 0 , which, starting from n > n 0 , all members of the sequence are located inside the interval ( A – , A+ ). A sequence having a limit is called convergent; otherwise - divergent.

A number sequence can have only one limit (finite or infinite) of a certain sign.

Example 5 . Harmonic sequence has the limit number 0. Indeed, for any interval (–; +) as a number N 0 can be any integer greater than . Then for everyone n > n 0 >we have

Example 6 . The sequence 2, 5, 2, 5,  is divergent. Indeed, no interval of length less than, for example, one, can contain all members of the sequence, starting from a certain number.

The sequence is called limited, if such a number exists M, What
for all n. Every convergent sequence is bounded. Every monotonic and bounded sequence has a limit. Every convergent sequence has a unique limit.