Numerical sequence. Designated numerical sequence Classification of numerical sequences and above them

If a function is assigned to the set of natural numbers N, then such a function is called an inexhaustible numerical sequence. Sound the numerical sequence means like (Xn), de n lie in the plurality of natural numbers N.

The numerical sequence can be given by a formula. For example, Xn=1/(2*n). In this order, we put the sing element of the sequence (Xn) in relation to the skin natural number n.

So now take n equals in sequence 1,2,3, …, we take the sequence (Xn): ½, ¼, 1/6, …, 1/(2*n), …

See the sequence

The succession can be fringed or not fringed, growing or decaying.

Sequence (Xn) name obmezhenoyu, to base two numbers m and M in such a way that for any n there are multiple natural numbers, the equality of m will be counted<=Xn

Sequence (Xn), yak not fringed, is called an uncircumcised succession.

growing, Like all natural numbers n, such equality X(n+1) > Xn wins. In other words, the skin member of the sequence, starting from another, is guilty of the greater one for the previous member.

Sequence (Xn) is called recessionary, for all natural numbers n such equality X(n+1) wins< Xn. Иначе говоря, каждый член последовательности, начиная со второго, должен быть меньше предыдущего члена.

An example of sequence

It is reversible that the sequences are 1/n and (n-1)/n successive.

If the sequence is recessive, then X(n+1)< Xn. Следовательно X(n+1) - Xn < 0.

X(n+1) - Xn = 1/(n+1) - 1/n = -1/(n*(n+1))< 0. Значит последовательность 1/n убывающая.

(n-1)/n:

X(n+1) - Xn =n/(n+1) - (n-1)/n = 1/(n*(n+1)) > 0. The sequence (n-1)/n is increasing.

Numerical sequence a numerical function is called, assigned to impersonal natural numbers .

How to set a function on impersonal natural numbers
, then the impersonal value of the function will be personal and skin number
be set as a valid number
. And here it seems that it is given numerical sequence. Call numbers elements but the members of the sequence, and the number - fire up abo -M Member of the sequence. Leather element maє advancing element
. I explain the use of the term "consequence".

Set the sequence by calling either the redistribution of її elements, or the law designated by which the element with the number is calculated , then. introductory formula її th member .

butt.Sequence
can be given by the formula:
.

Call the sequences as follows: etc., where the formula is indicated in the arches th member.

butt.Sequence
‑tse sequence

Anonymity of all elements of sequence
be appointed
.

Come on
і
- two sequences.

W ummah sequences
і
name the sequence
, de
, then.

R aznistyu these sequences are called sequence
, de
, then.

Yakscho і ‑ postiyni, then sequence
,
name linear combination sequences
і
, then.

Tvorom sequences
і
name the sequence -th member
, then.
.

Yakscho
, then you can privately
.

Suma, retail, TV and private sequence
і
are called їх algebraiccompositions.

butt.Let's look at the sequence
і
, De. Todi
, then. succession
maє all elements that are equal to zero.

,
, then. all the elements of creating a private equal
.

Yakshcho vikresli deyakі elements of sequence
so, if the impersonal elements are lost, then we take away another sequence, I call subsequence sequence
. Yakshcho vikresliti kіlka first elements in sequence
, then the new sequence is called too much.

Sequence
fencedbeast(bottom), which is impersonal
surrounded by the beast (from below). Name sequence obmezhenoyu like it is surrounded by the beast and below. The sequence is surrounded by the same or less than the other, if there are too many surpluses.

Similar sequences

Say what succession
converge, as a clear number so what for be-whom
issuing
, what for be-whom
, vykonuetsya nerіvnіst:
.

Number name boundary sequence
. When to write down
or
.

butt.
.

Let's show what
. Let's be a number
. Nerіvnіst
win for
, such that
, what is the designation for the number
. To mean,
.

In other words
means that all members of the sequence
with great numbers , then. starting from the current number
(when) sequence elements are in the interval
, which is called - Navkoro points .

Sequence
, between which is equal to zero (
, or
at
) is called infinitely small.

There are infinitely small fair assertions:

The sum of two infinitely small is infinitely small;

Tvіr infinitely small by the size of the obmezhena є infinitely small.

Theorem .In order for the consistency
small between, necessary and sufficient
, de - Postiyna; - extremely small .

The main power sequences that converge:

Powerfulness 3. and 4. zagalnyuyutsya on the verge of any number of sequences that converge.

Significantly, when counting between fractions, numerals and banners of such a linear combination of steps , between the fractions of the older members (that is, the members that avenge the greatest numeral and banner).

Sequence
called:

Mustache such sequences to call monotone.

Theorem . Like a sequence
monotonously grown and bordered by the beast, it converges and її between the old and її exact upper edges; as the sequence diminishes and is circumscribed from below, it converges to its exact lower edge.

Lecture 8. Number sequences.

Appointment8.1. As a skin meaning, it is necessary to put the speech number behind the sing lawx n , then impersonal enumeration of speech numbers

– abbreviations
, (8.1)

we will namenumerical sequence or just a consequence.

Okremi numbers x n  elements or members of sequence (8.1).

The sequence can be given by the formula of the capital term, for example:
or
. The sequence can be set ambiguously, for example, the sequence -1, 1, -1, 1, ... can be set by the formula
or
. Another way to win a recursive method for setting the sequence: the first number of members in the sequence is given and the formula for calculating the next elements. For example, sequence, which is designated as the first element and recurrent sequences
(arithmetic progression). Let's look at the sequence, as it is called Fibonacci order: the first two elements are given x 1 =1, x 2 = 1
for whatever
. We take the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, .... For whom it is important to know the formula of a hardened member.

8.1. Arithmetic dії іz sequences.

Let's look at two sequences:

(8.1)

Appointment 8.2. Nazvemothe work of succession
per number msuccession
. Let's write it like this:
.

We call the sequence sum of sequences (8.1) and (8.2) can be written as follows: ; similarly
called difference of sequences (8.1) and (8.2);
 the creation of sequences (8.1) and (8.2);  private sequences (8.1) and (8.2) (all elements
).

8.2. Exchanged and uncircumcised sequences.

Succulence of all elements of sufficient sequence
I make a simple numerical multiplicity, as if it could be surrounded by the beast (from below) and for some just purpose, similarly introduced for real numbers.

Appointment 8.3. Sequence
calledfouled beast , yakscho; M  top edge.

Appointment 8.4. Sequence
calledfringed below , yakscho;m  bottom edge.

Appointment 8.5.Sequence
calledobmezhenoyu that it is surrounded by the beast, and from below, so that it is possible to establish two speech numbers M andm so, that the skin element of sequence
satisfied with the inconsistencies:

, (8.3)

mіM– lower and upper faces
.

Irregularities (8.3) name mind obmezhennosti sequence
.

For example, sequence
swamped, and
uncircumscribed.

♦ Confirmation 8.1.
є obmezhenoyu .

Proof. Viberemo
. Deadline until appointment 8.5 sequence
you will be pissed off. ■

Appointment 8.6. Sequence
calledunrestricted like for any positive (like forever great) speech number A, there would like to be one element of sequencex n that satisfies the nervousness:
.

For example, sequence 1, 2, 1, 4, ..., 1, 2 n, … fringed at the bottom.

8.3. Infinitely great and infinitely small sequences.

Appointment 8.7. Sequence
calledinfinitely great , so for whatever (forever great) speech number A there is a number
such that for all
elementsx n
.

☼ Respect 8.1. Like the sequence is infinitely great, it is not bordered. And there is no trace of thinking that whether the sequence is unbounded and infinitely great. For example, sequence
not fenced, but unskinnedly great, because umova
don't win for all guys n. ☼

 Example 8.1.
є inexorably great. Let's take a number A>0. Nerіvnostі
acceptable n>A. Yakscho take
, then for all n>N you will overcome the nervousness
, so it is necessary for appointments 8.7, sequence
immeasurably large. 

Appointment 8.8. Sequence
calledinfinitely small , as for
(skіlki zavgodno small ) find number
such that for all
elements this sequence satisfies the nervousness
.

 Example 8.2. Let us know that the sequence inexorably small.

Let's take a number
. Nerіvnostі
acceptable . Yakscho take
, then for all n>N you will overcome the nervousness
. 

♦ Confirmation 8.2. Sequence
є infinitely great at
and inexorably small at
.

Proof.

1) Speak on the cob
:
, de
. Behind the Bernoulli formula (appendix 6.3, paragraph 6.1)
. Fixed rather positive number A and select a number for him N such that nerіvnіst was fair:

,
,
,
.

so yak
, then for the power of additional speech numbers for all

.

In this way, for
find such a number
, what for all


- infinitely large at
.

2) We look at the vipadok
,
(at q=0 may be trivial).

Come on
, de
, following the Bernoulli formula
or
.

Fixed
,
i vibero
like that

,
,
.

For

. Please enter this number N, what for all

, then at
succession
inexorably small. ■

8.4. The main power of infinitely small sequences.

♦ Theorem 8.1.Suma

і

Proof. Fixed ;
- extremely small

,

- extremely small

. Viberemo
. Todi at

,
,
. ■

♦ Theorem 8.2. Retail
two infinitely small sequences
і
є infinitely small sequence.

For prove Theorem to finish vikoristovuvaty nerіvnіst. ■

Last.The algebraic sum of any final number of infinitely small sequences is an infinitely small sequence.

♦ Theorem 8.3.Dobutok zamezhenoї sledovnіstі on neskіchenno small sledovnіst є neskіchenno ї skolіdovnіstі.

Proof.
- fried
- The sequence is infinitely small. Fixed ;
,
;
: at
fair
. Todi
. ■

♦ Theorem 8.4.Whether it be an infinitely small sequence є obmezhenoyu.

Proof. Fixed Come on, sprat. Todi
for all rooms n which means the exchange of succession. ■

Last. Tvіr two (and whether it be the end of the day) infinitely small sequences є infinitely small sequence.

♦ Theorem 8.5.

Like all the elements of infinitely small sequence
equal to the same numberc, then h = 0.

proof theorems to be carried out by the method
. ■

♦ Theorem 8.6. 1) Yakscho
- the sequence is infinitely great, then starting from the current numbern, marked private two sequences
і
, which is an infinitely small sequence.

2) Like all the elements of infinitely small sequence
vіdminnі vіd zero, then private two sequences
і
є inexorably great succession.

Proof.

1) Come on
- The sequence is immensely great. Fixed ;
or
at
. In this rank, for appointments 8.8 sequence - Immensely small.

2) Come on
- The sequence is infinitely small. It is acceptable that all the elements
vіdminnі vіd zero. Fixed A;
or
at
. For appointment 8.7 sequence immeasurably large. ■

As for the skin natural number n, the number x is assigned numerical sequence

x 1 , x 2 , … x n , …

Number x 1 call a member of the sequence with number 1 or the first member of the sequence, number x 2 - a member of the sequence with number 2 or another member of the sequence, etc. Number x n name member of sequence with number n.

Establish two ways of setting numerical sequences - for help and for help recurrent formula.

Order of the sequence for help formulas of the main member of the sequence- Tse zavdannya sequence

x 1 , x 2 , … x n , …

for an additional formula that shows the presence of a member x n in the form of a number n .

Example 1. Numerical sequence

1, 4, 9, … n 2 , …

given by the help formula of the captive term

x n = n 2 , n = 1, 2, 3, …

The order of the sequence for the additional formula, which turns the member of the sequence x n through the members of the sequence with forward numbers, is called the task of the sequence for the help recurrent formula.

x 1 , x 2 , … x n , …

name growing succession, more anterior member.

In other words, for all n

x n + 1 >x n

3 . Sequence of natural numbers

1, 2, 3, … n, …

є growing succession.

Appointment 2. Numerical sequence

x 1 , x 2 , … x n , …

name decaying succession, like a leather member of the chain of sequence less anterior member.

In other words, for all n= 1, 2, 3, …

x n + 1 < x n

Example 4. Sequence

given by the formula

є recessive succession.

Example 5 . Numerical sequence

1, - 1, 1, - 1, …

given by the formula

x n = (- 1) n , n = 1, 2, 3, …

not є neither growing nor falling succession.

Appointment 3. Increasing and decreasing number sequences are called monotonous sequences.

Exchange and non-exchange sequence

Appointment 4. Numerical sequence

x 1 , x 2 , … x n , …

name fringed beast, as such, the number M, the skin member of the sequence less M numbers.

In other words, for all n= 1, 2, 3, …

Appointment 5. Numerical sequence

x 1 , x 2 , … x n , …

name fringed below, yakscho іsnuє taka kіlkіst m, skin member of the qєї sequence more number m.

In other words, for all n= 1, 2, 3, …

Appointment 6. Numerical sequence

x 1 , x 2 , … x n , …

called obmezhenoyu, like out fenced and burned, and from below.

In other words, to base such numbers M and m, which for all n= 1, 2, 3, …

m< x n < M

Appointment 7. Numbers of sequences, yaks not є obezhenimi, name irrelevant sequences.

Example 6 . Numerical sequence

1, 4, 9, … n 2 , …

given by the formula

x n = n 2 , n = 1, 2, 3, … ,

bordered at the bottom for example, the number 0. However, the sequence not surrounded by the beast.

Example 7. Sequence

.

Mathematics is a science, as a future world. Like an opinion, so simple is a person - no one can do without her. Start a bunch of little children to rahuvat, then fold, see, multiply and extend, until the middle school, letters of designation enter into the head, and the elder cannot do without them.

But today we will talk about those on which all mathematics will be based. About the grouping of numbers under the name "between sequences".

What is the sequence and de їhnya boundary?

The meaning of the word "consequence" is not very important to interpret. Tse taka pobudova speeches, dehto schos schos rastashovaniya in the singing order chi cherzі. For example, a ticket for receipts to the zoo is the sequence. Moreover, there might be more than one! Like, for example, marvel at the drawing in the store, there is only one sequence. And if one person s tsієї cherga raptom pіde, then tse even іnsha cherga, іnsha way.

The word "border" is also easy to interpret - tse end of chogos. However, the mathematics of intersequences has the same meaning on the number line, such as the correct sequence of numbers. Why pragne, but chi does not end? Everything is simple, there is no point in the number line, and the greater number of sequences, like a change, can only spin the cob and look like this:

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

Zvіdsi vyznachennya sequentіє є funktієyu natural argument. In simpler words - a number of members of the deaky multiplier.

How will the numerical sequence be?

The simplest example of numerical sequence can look like this: 1, 2, 3, 4, …n…

Most often, for practical purposes, the sequence will be from numbers, moreover, the skin of the offensive member is low, significantly X, may own name. For example:

x 1 - the first member of the sequence;

x 2 - another member of the sequence;

x 3 - the third member;

x n - th member.

In practical methods, the sequence is given by a bold formula, in others it is changed. For example:

X n \u003d 3n, then the series of numbers itself looks like this:

Varto do not forget that when recording sequences in a flashy way, one can sing whether they are Latin letters, and not only X. For example: y, z, k, etc.

Arithmetic progression as a part of sequences

The first lower shukati intersequences, dotsilno more easily erroneous at the very understanding of a similar numerical series, with which they stumbled, being the middle classes. An arithmetic progression is a series of numbers, in which the difference between the judicial members has become.

Task: “Let's go and 1 = 15, and the progress of the numerical low is d = 4. Stay the first 4 members of the row "

Solution: a 1 = 15 (behind the mind) - the first member of the progression (number series).

and 2 \u003d 15 + 4 \u003d 19 is the other term of the progression.

and 3 \u003d 19 + 4 \u003d 23 is the third term.

and 4 \u003d 23 + 4 \u003d 27 is the fourth term.

However, let's use the method to find great values, for example, up to a 125. . Specially for such vipadkіv, a formula was created that is easy for practice: a n \u003d a 1 + d (n-1). In times a 125 =15+4(125-1)=511.

See sequences

The greater number of sequences is inexcusable; Іsnuє two tsіkavі see the numerical series. The first one is given by the formula a n = (-1) n. Mathematicians often call this sequence a flasher. Why? Let's revisit the number series.

1, 1, -1, 1, -1, 1 and so on. On a similar example, it becomes clear that the numbers in sequences can be easily repeated.

Factorial sequence. It is easy to guess - the formula that sets the sequence has a factorial. For example: and n = (n+1)!

Then the sequence will look like this:

and 2 \u003d 1x2x3 \u003d 6;

and 3 \u003d 1x2x3x4 \u003d 24 and so on.

Sequence, given by arithmetic progression, is called inexorably recessive, as all of its members are subject to unevenness -1

and 3 \u003d - 1/8 thinly.

Іsnuє navit sledovnіst, scho folded from one and the same number. So, and n \u003d 6 is added up with an infinite number of sixths.

Appointment between sequences

Between sequences have long been explored in mathematics. Zvichayno, the stench deserved to be competently designed in their own right. Otzhe, it's time to find out about the appointment between sequences. For the cob, let's take a look at the boundary for the linear function:

1. Mustache between are indicated shortly lim.
2. The record of the boundary is formed from the shortness of lim, whether it be changed, which is right up to the singing number, zero or inconsistency, and also from the function itself.

It is easy to understand that the designation of inter-sequence can be formulated as follows: it is the number to which all the members of the sequence are infinitely approaching. Simple butt: x = 4x+1. Then the sequence itself will look like this.

5, 9, 13, 17, 21...x...

In this order, the sequence is given to inexorably increase, and, therefore, the intermediary inconsistency at x→∞, and write down the following as follows:

Well, take the following sequence, but if you will bend them to 1, then take it away:

And the series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. It is necessary for Schoraz to present the number of daedalus closer to one (0.1, 0.2, 0.9, 0.986). From the 3rd row it is clear that there are five five functions between them.

From the point of part of the varto memorize, what is the inter-numerical sequence, the method of solving simple problems is designated.

Significant signification between sequences

Razіbravshi between the numerical sequence, the designation of yoga and apply, you can proceed to folding topics. Absolutely all inter-sequences can be formulated with one formula, as you call it in the first semester.

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

∀ is a verbal quantifier, which replaces phrases for everyone, for everything, etc.

∃ is the quantifier of the meaning, at the time it means that the value of N is known, that the impersonal natural numbers lie.

Dovga is a vertical stick, which goes behind N, which means that it is impersonal N “so, sho”. Indeed, it can mean “so, so”, “so, so” too.

To fix the material, read the formula out loud.

Insignificance and significance between

The method of understanding between sequences, which, when looked at more, is simple and simple in practice, but not so rational in practice. Try to know the boundary for this function:

If you represent the different values ​​of “ix” (with the skin at a time increase: 10, 100, 1000 and so on), then in the number book we take ∞, but in the banner tezh ∞. Go out to finish the wonderful drib:

Ale chi tse so true? Calculate the inter-numerical sequence at the time you can do it easily. It would be possible to get rid of everything, like, even if you are ready, and take it out on reasonable minds, but there is one more way especially for such vipadkivs.

For the cob, we know the senior step of the fractional numerator - tse 1, so that x can be shown like x 1.

Now we know the senior step at the bannerman. Tag 1.

Let's send a number book, and a banner to change the most beautiful world. In times, fractions are divisible by x 1.

Let us know what is the significance of pragne skin dodanok, what to avenge the change. Fractions are seen at a glance. At x→∞, the value of skin fraction is zero. When executing the work in a written form, you should create the following wines:

Come out viraz:

Zvichayno, the fractions that avenge x did not become zeros! And yet, the meaning of the flooring is not enough, so it is possible not to protect yoga with roses. True, x will not be equal to 0 in this case, even if it is not possible to divide by zero.

What is the neighborhood?

Admittedly, the order of the professor has a coherent sequence, obviously set by the smallest folding formula. Professor knows how to walk Adzhe all people have mercy.

Auguste Cauchy, in his own time, foresaw the best way to bring between sequences. This method was called the operation of the outskirts.

Let's assume that there is a point a, її near the edge on the numerical straight line ε ("epsilon"). The remains of the rest are changed - come on, її the meaning is always positive.

Now let's put a succession x n і let's put the tenth member of the sequence (x 10) into the neighborhood of a. How can I write down this fact in my mathematical language?

It is acceptable that x 10 be right in the same point as the same as in 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 the hour has come to explain in a practical way that formula, about which it was said more. Indeed, the number a is rightly called the end point of the sequence, as for whether it is between the two, the unevenness ε>0 is fixed, moreover, the whole neighborhood may have its own natural number N, such that all members of the sequence with the most significant numbers appear in the middle of the sequence x n - a |< ε.

With such knowledge, it is easy to create a variance between sequences, to bring it to a ready conclusion.

Teoremi

Theories about intersequences are important warehouse theory, without which practice is impossible. If you remember more than a couple of heady theorems, you can make it easier to prove the solution:

1. Unity between sequences. The boundary in the be-yakіy sequence can be only one or not be zovsіm. That very butt from black, which may have only one end.
2. If a series of numbers is between, then the sequence of these numbers is separated.
3. Between the sumi (retail, create) sequences of the costly sum (retail, creation) and inter.
4. The boundary between the private view of the two sequences is more expensive than the private boundary between the same and only the same, if the banner does not turn to zero.

Proof of sequences

Sometimes I need to turn around the task, bring the task between numerical sequences. Let's look at an example.

To bring, scho between sequences, given by the formula, to zero.

Behind a higher rule, for the sake of consistency, there may be nervousness | x n - a |<ε. Подставим заданное значение и точку отсчёта. Получим:

Virazimo n through "epsilon", to show the reason of any number and to bring the obviousness of the inter-sequence.

At which stage it is important to guess that "epsilon" and "en" - the numbers are positive and do not equal zero. Now you can continue further transformation, victorious knowledge about the nervousness, taken away from the middle school.

Sounds like n> -3 + 1/ε. If you think about the natural numbers, then the result can be rounded off by adding the yogo to the square bow. In this manner, it was brought to light that for any value near the “epsilon” point a = 0, there was such a thing that there was a lot of unevenness. You can boldly affirm that the number is between the given sequence. What did it take to bring.

The axis by such an easy method can be brought between numerical sequences, which would not be foldable at first glance. Golovna - do not panic, after succumbing to the task.

Or maybe no yoga?

The basis of intersequence is neobov'yazkovo practical. It is easy to understand such a series of numbers, as if it were true not to think of the end. For example, that “flasher” itself x n = (-1) n. I realized that the sequence, which is composed of less than two numbers, cyclically repeating, is impossible for the mother between.

That very history repeats itself with sequences that add up from one number, fractional ones, which can be calculated in the course of non-significance in any order (0/0, ∞/∞, ∞/0 just). Keep in mind that the calculation may not be correct. Sometimes between the sequences, you can help to re-verify your decision.

Monotonous sequence

More often, we looked at a few examples of sequences, methods of their perfection, and now let's try to take a sing-song note and call it “monotonous sequence”.

Appointment: whether the sequence is rightly called monotonously growing, as if for it the suvor unevenness is victorious x n< x n +1. Также любую последовательность справедливо называть монотонной убывающей, если для неё выполняется неравенство x n >xn+1.

The order of these two minds is similar to the same inconsistency. Obviously, x n ≤ x n +1 (increasing sequence) і x n ≥ x n +1 (non-increasing sequence).

Ale is easier to understand on the butts.

The sequence, given by the formula x n = 2 + n, makes the next series of numbers: 4, 5, 6 thin. bud. This is a monotonously growing sequence.

And if you take x n \u003d 1 / n, then we take the row: 1/3, ¼, 1/5 and so on.

Boundary of similarity and boundary sequence

The sequence is obmezhena - the sequence that can be between. The sequence to go is a series of numbers that can be infinitely small between.

In this order, between the interchangeable sequences - tse be-yak deisne chi complex number. Remember that there may be more than one boundary.

Between the sequences, which zbіgaєtsya - tse value is infinitely small (dіysna or complex). If you crucify the diagram of sequence, then the singing points will converge, bend and turn into a singing magnitude. Zvіdsi i name - sequence, scho zbіgaєtsya.

Between the monotonous sequence

Between such a sequence, you may be, or you may not be. On the back of your mind, if you have wine, you can look for evidence of the presence of the boundary.

In the midst of monotonous sequences, I see I go and rozbіzhnu. Skhidna - tse such a sequence, as it was established impersonal x and may in this multiplicity of actions or a complex boundary. Razbіzhna - sequence, which cannot be between its own plurality (neither deysnoy, nor complex).

Moreover, the sequence converge, as if from a geometric image, the upper and lower boundaries converge.

The boundary of similar sequences in rich vipads can be equal to zero, so if there is an infinitely small sequence of may visible boundary (zero).

Yaku sequence, scho to go, do not take, all the stench of the environment, prote far from all the environment of the sequence converge.

Sum, retail, dobutok two sequences that converge - the sequence is also similar. However, privately it can also be similar, as it is marked out!

Rіznі dії z between the borders

Between sequences - the same value (in most cases) value, like digits and numbers: 1, 2, 15, 24, 362 and so on.

First, like the digits of that number, between any sequences you can add up and see. Vykhodyachi z third theorem about between sequences, such equality is true: between the sum of sequences is more than the sum of their inter.

In another way, based on the fourth theorem about inter sequences, the following equality is true: between dobutku n-th number of sequences dobutku їх inter. The same is true for the division: the boundary between a private two sequences is equal to a private one between them, for the mind that a boundary is not equal to zero. Even if between the sequences is equal to zero, then we have gone to zero, which is impossible.

The power of the magnitudes of the sequences

It would have been better if, between the numerical sequence, it was already decided to report it, but such phrases are repeatedly guessed, like “infinitely small” and “infinitely large” numbers. Obviously, if it is a sequence of 1/x, de x→∞, then such a drop is infinitely small, and if it is the same sequence, even if it is between zero (x → 0), then the drop becomes an infinitely large value. And such values ​​may have their own peculiarities. The dominance of inter-sequence, which may be small or large, is similar to the offensive:

1. The sum, whether it be a small number of small quantities, will always be a small amount.
2. The sum of any number of great magnitudes will be an infinitely great magnitude.
3. Tvіr like always small quantities is infinitely small.
4. Dobutok skіlki zavgodno great numbers - the value is infinitely greater.
5. If the succession of an infinitely great number is visible, then the value that is reversible will be infinitely small and go to zero.

Accurately calculate between sequences - not such a coherent task, if you know a simple algorithm. Ale between sequences is a topic that requires maximum respect and perseverance. Zvichayno, just to catch the essence of the manifestation of similar viruses. Starting small, with a year you can reach great heights.