For any monotonic bounded sequence exists. Number sequences

Definition 1. A sequence is called non-decreasing [non-increasing] if each element of the sequence, starting from the second, is not less than [not more than] its previous element, that is, if the inequality is true for all numbers

Definition 2. A sequence is called monotonic if it is either non-decreasing or non-increasing.

If the elements of a non-decreasing sequence for all numbers satisfy a strict inequality, then this sequence is called increasing.

Similarly, if the elements of a non-increasing sequence for all numbers satisfy a strict inequality, then this sequence is called decreasing.

Note that every monotonic sequence is obviously bounded on one side (either from above or from below). Indeed, every non-decreasing sequence is bounded from below (the value of its first element can be taken as the lower bound), and every non-increasing sequence is bounded above (the value of its first element can also be taken as the upper bound).

It follows that a non-decreasing sequence will be bounded on both sides, or simply bounded, if and only if it is bounded above, and a non-increasing sequence will be bounded if and only if it is bounded below.

Let's look at examples of monotonic sequences.

1. The sequence is non-decreasing. It is limited from below by the value of its first element, but is not limited from above.

2. The sequence is decreasing. It is limited on both sides: from above by the value of its first element 2, and from below, for example, by the number 1.

Definition. The sequence (x n) is called limited, if there is a number M>0 such that for any n the inequality is true:

those. all members of the sequence belong to the interval (-M; M).

For example, the sequences 2 0), 3 0), 4 0), 5 0) are limited, and the sequence 1 0) is unlimited.

The theorem follows directly from the definition of a bounded sequence and the definition of the limit of a sequence:

Theorem. If x n ® a, then the sequence (x n ) is bounded.

It should be noted that the converse statement is not true, i.e. the boundedness of a sequence does not imply its convergence.

For example, the sequence has no limit though

Definition. The sequence (x n) is called bounded above, if for any n there is a number M such that x n £ M.

Example.(x n ) = 3n – bounded below (3, 6, 9, …).

Monotonous sequences.

Definition. 1) If x n +1 > x n for all n, then the sequence is increasing.

2) If x n +1 ³ x n for all n, then the sequence is non-decreasing.

3) If x n +1< x n для всех n, то последовательность убывающая.

4)If x n +1 £ x n for all n, then the sequence is non-increasing

All these sequences are called monotonous. Increasing and decreasing sequences are called strictly monotonous.

Example.(x n ) = 1/n – decreasing and limited

(x n ) = n – increasing and unlimited.

Example. Prove that the sequence (x n )= is monotonic increasing.

Solution. Let's find a member of the sequence (x n +1 )=

Let's find the sign of the difference: (x n)-(x n +1)=

, because nÎN, then the denominator is positive for any n.

Thus, x n +1 > x n . The sequence is increasing, which should have been proven.

Example. Find out whether the sequence is increasing or decreasing

Solution. Let's find it. Let's find the difference

Because nÎN, then 1 – 4n<0, т.е. х n+1 < x n . Последовательность монотонно убывает.

It should be noted that monotonic sequences are limited on at least one side.

Theorem. A monotonic bounded sequence has a limit.

Proof. Consider a monotonic non-decreasing sequence

x 1 £ x 2 £ x 3 £ … £ x n £ x n +1 £ …

This sequence is bounded from above: x n £ M, where M is a certain number.

Because Any numerical set bounded above has a clear upper bound, then for any e>0 there is a number N such that x N > a - e, where a is some upper bound of the set.

Because (x n) is a non-decreasing sequence, then for N > n a - e< x N £ x n ,

Hence a - e< x n < a + e

E< x n – a < e или ôx n - aô< e, т.е. lim x n = a.

For other monotonic sequences the proof is similar.

The theorem has been proven.

§3. Number e.

Consider the sequence (x n ) = .

If the sequence (x n) is monotonic and bounded, then it has a finite limit.

According to Newton's binomial formula:

Or what's the same

Let us show that the sequence (x n) is increasing. Indeed, let’s write down the expression x n +1 and compare it with the expression x n:

Each term in the expression x n +1 is greater than the corresponding value x n, and, in addition, x n +1 has one more positive term added. Thus, the sequence (x n) is increasing.

Let us now prove that for any n its terms do not exceed three: x n< 3.

So, the sequence is monotonically increasing and bounded from above, i.e. has a finite limit. This limit is usually denoted by the letter e.

From the inequality it follows that e £ 3. Discarding all terms in the equality for (x n), starting from the fourth, we have:

passing to the limit, we get

Thus, the number e is contained between the numbers 2.5 and 3. If you take more terms of the series, you can get a more accurate estimate of the value of the number e.

It can be shown that the number e is irrational and its value is 2.71828...

Similarly, it can be shown that , expanding the requirements for x to any real number:

Let's assume:

The number e is the base of the natural logarithm.

Above is the graph of the function y = lnx.

Relationship between natural and decimal logarithms.

Let x = 10 y, then lnx = ln10 y, therefore lnx = yln10

y = , where M = 1/ln10 » 0.43429… is the transition module.

§4. The concept of the limit of a function.

4.1. Limit of a function at a point.

y f(x)

0 a - D a a + D x

Let the function f(x) be defined in a certain neighborhood of the point x = a (i.e., at the point x = a the function may not be defined)

Definition. The number A is called limit functions f(x) for x®a, if for any e>0 there is a number D>0 such that for all x such that

ïx - aï< D

the inequality ïf(x) - Aï is true< e.

The same definition can be written in another form:

If a - D< x < a + D, x ¹ a, то верно неравенство А - e < f(x) < A + e.

Writing the limit of a function at a point:

Basic theorems about limits.

Theorem 1. , where C = const.

The following theorems are valid under the assumption that the functions f(x) and g(x) have finite limits for x®a.

Theorem 2.

The proof of this theorem will be given below.

Theorem 3.

Consequence.

Theorem 4. at

Theorem 5. If f(x)>0 near the point x = a and , then A>0.

The sign of the limit at f(x) is determined similarly< 0, f(x) ³ 0, f(x) £ 0.

Theorem 6. If g(x) £ f(x) £ u(x) near the point x = a and , then and .

Definition. The function f(x) is called limited near the point x = a, if there is a number M>0 such that ïf(x)ï

Theorem 7. If the function f(x) has a finite limit at x®a, then it is limited near the point x = a.

Proof. Let , i.e. , Then

Where M = e + ïАï

The theorem has been proven.

4.2. One-sided limits.

Definition. If f(x) ® A 1 at x ® a only at x< a, то - называется limit function f(x) at point x = a left, and if f(x) ® A 2 for x ® a only for x > a, then called limit function f(x) at point x = a on right.

at

The above definition refers to the case when the function f(x) is not defined at the point x = a itself, but is defined in some arbitrarily small neighborhood of this point.

Limits A 1 and A 2 are also called one-way limits function f(x) at point x = a. It is also said that A - final limit functions f(x).

4.3.The limit of a function as the argument tends to infinity.

Definition. The number A is called limit function f(x) for x®¥, if for any number e>0 there is a number M>0 such that for all x, ïxï>M the inequality holds

Definition 1. The sequence is called decreasing (non-increasing ), if for everyone
inequality holds
.

Definition 2. Consistency
called increasing (non-decreasing ), if for everyone
inequality holds
.

Definition 3. Decreasing, non-increasing, increasing and non-decreasing sequences are called monotonous sequences, decreasing and increasing sequences are also called strictly monotonous sequences.

Obviously, a non-decreasing sequence is bounded from below, and a non-increasing sequence is bounded from above. Therefore, any monotonic sequence is obviously limited on one side.

Example 1. Consistency
increases, does not decrease,
decreases
does not increase
– non-monotonic sequence.

For monotonic sequences, the following plays an important role:

Theorem 1. If a nondecreasing (nonincreasing) sequence is bounded above (below), then it converges.

Proof. Let the sequence
does not decrease and is bounded from above, i.e.
and many
limited from above. By Theorem 1 § 2 there is
. Let's prove that
.

Let's take
arbitrarily. Because the A– exact upper bound, there is a number N such that
. Since the sequence is non-decreasing, then for all
we have, i.e.
, That's why
for all
, and this means that
.

For a nonincreasing sequence bounded below, the proof is similar to ( students can prove this statement at home on their own). The theorem has been proven.

Comment. Theorem 1 can be formulated differently.

Theorem 2. In order for a monotonic sequence to converge, it is necessary and sufficient that it be bounded.

Sufficiency is established in Theorem 1, necessity – in Theorem 2 of § 5.

The monotonicity condition is not necessary for the convergence of a sequence, since a convergent sequence is not necessarily monotonic. For example, the sequence
not monotonic, but converges to zero.

Consequence. If the sequence
increases (decreases) and is limited from above (from below), then
(
).

Indeed, by Theorem 1
(
).

Definition 4. If
at
, then the sequence is called contracting system of nested segments .

Theorem 3 (principle of nested segments). Every contracting system of nested segments has a unique point With, belonging to all segments of this system.

Proof. Let us prove that the point With exists. Because the
, That
and therefore the sequence
does not decrease, but the sequence
does not increase. Wherein
And
limited because. Then, by Theorem 1, there exist
And
, but since
, That
=
. Found point With belongs to all segments of the system, since by the corollary of Theorem 1
,
, i.e.
for all values n.

Let us now show that the point With- the only one. Let's assume that there are two such points: With And d and let for certainty
. Then the segment
belongs to all segments
, i.e.
for all n, which is impossible, since
and, therefore, starting from a certain number,
. The theorem has been proven.

Note that the essential thing here is that closed intervals are considered, i.e. segments. If we consider a system of contracting intervals, then the principle is, generally speaking, incorrect. For example, intervals
, obviously contract to a point
, however point
does not belong to any interval of this system.

Let us now consider examples of convergent monotonic sequences.

1) Number e.

Let us now consider the sequence
. How is she behaving? Base

degrees
, That's why
? On the other side,
, A
, That's why
? Or is there no limit?

To answer these questions, consider the auxiliary sequence
. Let us prove that it decreases and is bounded below. At the same time, we will need

Lemma. If
, then for all natural values n we have

(Bernoulli's inequality).

Proof. Let's use the method of mathematical induction.

If
, That
, i.e. the inequality is true.

Let's assume that it is true for
and prove its validity for
+1.

Right
. Let's multiply this inequality by
:

Thus, . This means, according to the principle of mathematical induction, Bernoulli’s inequality is true for all natural values n. The lemma is proven.

Let us show that the sequence
decreases. We have

‌‌‌׀Bernoulli's inequality׀
, and this means that the sequence
decreases.

Boundedness from below follows from the inequality
‌‌‌׀Bernoulli's inequality׀
for all natural values n.

By Theorem 1 there is
, which is denoted by the letter e. That's why
.

Number e irrational and transcendental, e= 2.718281828… . It is, as is known, the base of natural logarithms.

Notes. 1) Bernoulli's inequality can be used to prove that
at
. Indeed, if
, That
. Then, according to Bernoulli’s inequality, with
. Hence, at
we have
, that is
at
.

2) In the example discussed above, the base of the degree tends to 1, and the exponent n- To , that is, there is uncertainty of the form . Uncertainty of this kind, as we have shown, is revealed by the remarkable limit
.

2)
(*)

Let us prove that this sequence converges. To do this, we show that it is bounded from below and does not increase. In this case, we use the inequality
for all
, which is a consequence of the inequality
.

We have
see inequality is higher
, i.e. the sequence is bounded below by the number
.

Further,
since

, i.e. the sequence does not increase.

By Theorem 1 there is
, which we denote X. Passing in equality (*) to the limit at
, we get

, i.e.
, where
(we take the plus sign, since all terms of the sequence are positive).

The sequence (*) is used in the calculation
approximately. Behind take any positive number. For example, let's find
. Let
. Then
,. Thus,
.

3)
.

We have
. Because the
at
, there is a number N, such that for everyone
inequality holds
. So the sequence
, starting from some number N, decreases and is bounded below, since
for all values n. This means that by Theorem 1 there is
. Because the
, we have
.

So,
.

4)
, on right - n roots

Using the method of mathematical induction we will show that
for all values n. We have
. Let
. Then, from here we obtain a statement based on the principle of mathematical induction. Using this fact, we find, i.e. subsequence
increases and is bounded from above. Therefore it exists because
.

Thus,
.

The elements of which do not decrease with increasing number, or, conversely, do not increase. Such sequences are often encountered in research and have a number of distinctive features and additional properties. A sequence of one number cannot be considered ascending or descending.

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  Let there be a set X (\displaystyle X), on which the order relation is introduced.

  Sequence of set elements X (\displaystyle X) called non-decreasing , if each element of this sequence is not greater than the next one.

  ( x n ) (\displaystyle \(x_(n)\))- non-decreasing ⇔ ∀ n ∈ N: x n ⩽ x n + 1 (\displaystyle \Leftrightarrow ~\forall n\in \mathbb (N) \colon x_(n)\leqslant x_(n+1))

  Subsequence ( x n ) (\displaystyle \(x_(n)\)) elements of the set X (\displaystyle X) called non-increasing , if each next element of this sequence does not exceed the previous one.

  ( x n ) (\displaystyle \(x_(n)\))- non-increasing ⇔ ∀ n ∈ N: x n ⩾ x n + 1 (\displaystyle \Leftrightarrow ~\forall n\in \mathbb (N) \colon x_(n)\geqslant x_(n+1))

  Subsequence ( x n ) (\displaystyle \(x_(n)\)) elements of the set X (\displaystyle X) called increasing , if each next element of this sequence is greater than the previous one.

  ( x n ) (\displaystyle \(x_(n)\))- increasing ⇔ ∀ n ∈ N: x n< x n + 1 {\displaystyle \Leftrightarrow ~\forall n\in \mathbb {N} \colon x_{n}

  Subsequence ( x n ) (\displaystyle \(x_(n)\)) elements of the set X (\displaystyle X) called decreasing , if each element of this sequence is greater than the next one.

  ( x n ) (\displaystyle \(x_(n)\))- decreasing ⇔ ∀ n ∈ N: x n > x n + 1 (\displaystyle \Leftrightarrow ~\forall n\in \mathbb (N) \colon x_(n)>x_(n+1))

  monotonous, if it is non-decreasing or non-increasing.

  The sequence is called strictly monotonous, if it is increasing or decreasing.

  Obviously, a strictly monotonic sequence is monotonic.

  Sometimes a variant of terminology is used in which the term "increasing sequence" is considered as a synonym for the term "non-decreasing sequence", and the term "decreasing sequence" is considered as a synonym for the term "non-increasing sequence". In such a case, the increasing and decreasing sequences from the above definition are called “strictly increasing” and “strictly decreasing”, respectively.

  Intervals of monotony

  It may turn out that the above conditions are not met for all numbers n ∈ N (\displaystyle n\in \mathbb (N) ), but only for numbers from a certain range

  I = ( n ∈ N ∣ N − ⩽ n< N + } {\displaystyle I=\{n\in \mathbb {N} \mid N_{-}\leqslant n

  (here it is allowed to reverse the right border N + (\displaystyle N_(+)) to infinity). In this case the sequence is called monotonic on the interval I (\displaystyle I) , and the range itself I (\displaystyle I) called an interval of monotony sequences.

  If each natural number n is associated with some real number x n, then we say that the given number sequence

  x 1 , x 2 , … x n , …

  Number x 1 is called a member of the sequence with number 1 or first term of the sequence, number x 2 - member of the sequence with number 2 or the second member of the sequence, etc. The number x n is called member of the sequence with number n.

  There are two ways to specify number sequences - with and with recurrent formula.

  Sequence using formulas for the general term of a sequence– this is a sequence task

  x 1 , x 2 , … x n , …

  using a formula expressing the dependence of the term x n on its number n.

  Example 1. Number sequence

  1, 4, 9, … n 2 , …

  given using the common term formula

  x n = n 2 , n = 1, 2, 3, …

  Specifying a sequence using a formula expressing a sequence member x n through the sequence members with preceding numbers is called specifying a sequence using recurrent formula.

  x 1 , x 2 , … x n , …

  called in increasing sequence, more previous member.

  In other words, for everyone n

  x n + 1 >x n

  Example 3. Sequence of natural numbers

  1, 2, 3, … n, …

  is ascending sequence.

  Definition 2. Number sequence

  x 1 , x 2 , … x n , …

  called descending sequence if each member of this sequence less previous member.

  In other words, for everyone n= 1, 2, 3, … the inequality is satisfied

  x n + 1 < x n

  Example 4. Subsequence

  

  given by the formula

  is descending sequence.

  Example 5. Number sequence

  1, - 1, 1, - 1, …

  given by the formula

  x n = (- 1) n , n = 1, 2, 3, …

  is not neither increasing nor decreasing sequence.

  Definition 3. Increasing and decreasing number sequences are called monotonic sequences.

  Bounded and Unbounded Sequences

  Definition 4. Number sequence

  x 1 , x 2 , … x n , …

  called limited from above, if there is a number M such that each member of this sequence less numbers M.

  In other words, for everyone n= 1, 2, 3, … the inequality is satisfied

  Definition 5. Number sequence

  x 1 , x 2 , … x n , …

  called bounded below, if there is a number m such that each member of this sequence more numbers m.

  In other words, for everyone n= 1, 2, 3, … the inequality is satisfied

  Definition 6. Number sequence

  x 1 , x 2 , … x n , …

  is called limited if it limited both above and below.

  In other words, there are numbers M and m such that for all n= 1, 2, 3, … the inequality is satisfied

  m< x n < M

  Definition 7. Numeric sequences that are not limited, called unlimited sequences.

  Example 6. Number sequence

  1, 4, 9, … n 2 , …

  given by the formula

  x n = n 2 , n = 1, 2, 3, … ,

  bounded below, for example, the number 0. However, this sequence unlimited from above.

  Example 7. Subsequence

  given by the formula

  is limited sequence, because for everyone n= 1, 2, 3, … the inequality is satisfied

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