Let some sequence of renumbered numbers x 1 , x 2 ,..., x n ,.. . be given, which we denote briefly or (x n ) . This sequence can be written as a function of the number n: x n =f(n) , or x 1 =f(1) , x 2 =f(2),.. ., x n =f(n),.. ..

Any sequence will be specified if the rule for the formation of its members is specified. The sequence is usually given by formulas like x n =f(n) or x n =f(x n-1) , x n =f(x n-1 , x n-2) etc., where .

Example.Sequence 2, 4, 8, 16, .. . given by the formula x n =2 n ; geometric progression a 1 , a 2 ,..., a n , .. . can be defined by the formula a n =a 1 q n-1 or a n =a n-1 q ; Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, .. . are defined by the formulas x n =x n-1 +x n-2 , n=3, 4, .. ., x 1 =1 , x 2 =1 .

Number Sequence Graph(x n ) is formed by a set of points M n (n;f(n)) on the nOx plane, i.e. number sequence chart consists of discrete points.

The sequence (x n ) is called increasing if the condition of the form is satisfied.

The sequence (x n ) is called decreasing if the condition of the form is satisfied.

The sequence (x n ) is called non-increasing if the condition of the form is satisfied.

The sequence (x n ) is called non-decreasing if the following condition is met: .

Such sequences are called monotonic. The remaining sequences are not monotonic.

Next is called endless sequence any objects of the same nature.

Example.Series of numbers - number series. Some of the functions - functional range.

The order of the elements of a series is significant. By changing the order, we get another row from the same elements.

We are only interested here in the numerical series and its sum, which is still written formally (not constructively, not formalized), that is, the sum of all members of some infinite numerical sequence u 1 , u 2 ,..., u n ,.. ., or u 1 + u 2 +...+u n +.. .. This series can be written compactly as

The sign is the "sigma" sign or the sign of the sum, the sequential summation of all elements u n from the lower limit n=1 (indicated at the bottom, can be either finite or negative infinity) to the upper limit (indicated at the top, can be any number, greater than or equal to the lower limit, as well as positive infinity).

The numbers u n (n=1, 2, .. .) are called members of the series, and u n is the common member of the series.

Example.In a school mathematics course, a geometric infinitely decreasing progression is given a=aq+aq 2 +...+aq n-1 +.. ., |q|<1 , u 1 =a , u 2 =aq, .. ., u n = aq n-1 . Сумма этого ряда (прогрессии), как известно из школьного курса, равна S=a/(1-q) .

Example. Harmonic series of numbers- series of the form: . Below we will consider it in more detail.

The number series will be considered given, that is, each of its elements will be uniquely determined if the rule for finding its common member is specified or some numeric function natural argument , or u n =f(n) .

Example.If , then the series is given , or in compact notation:

If given harmonic series of numbers, then its common term can be written as , and the series itself can be written as

Let us give the definition of a finite sum of a series and a sequence of such finite sums.

The final sum of the n first terms of the series is called its n-th partial sum and is denoted by S n :

This sum is found according to the usual rules for summing numbers. There are infinitely many such sums, that is, for each series, one can consider a series composed of partial sums: S 1 , S 2 ,... , S n , .. . or a sequence of partial sums constructed for this series: .

The sequence is bounded from above, if there is such a common number M for all members of the sequence, which is not exceeded by all members of the sequence, that is, if the following condition is satisfied:

The sequence of numbers is bounded from below, if there is a common number m for all members of the sequence, which exceeds all members of the sequence, that is, if the condition is met:

The sequence of numbers is limited if there are numbers m and M that are common to all members of the sequence and satisfy the condition:

The number a is called the limit of the numerical sequence(x n ) , if there is such a small number that all the members of the sequence, except for some finite number of first members, fall into the - neighborhood of the number a , that is, in the end, they condense around the point a . Thus, all points x i , i=N 0 , N 0 +1 , N 0 +2, .. must fall into the interval. sequences. In this case, the number N 0 depends on the chosen number, that is, (Fig. 7.1) .

Rice. 7.1.

Mathematically, the existence of a sequence limit can be written as:

This fact is written briefly as or , and say that it converges to the number a . If the sequence has no limit, then it is called divergent.

It follows directly from the definition of the limit: if we discard, add or change a finite number of members of the sequence, then the convergence is not violated (that is, if the original sequence converges, then the modified sequence converges) and the limits of the original and resulting sequences will be equal.

Example.Assume that , where , that is , , . This fact is easily proved, but for now we take it as a proven fact. Then , : . Find the value of the number (if such a number exists). Consider . The following relation is true:

So if we take a number , then the inequality will be satisfied. For example, with the value , we get the number N 0 =99 , that is, |x n -1|<0,01 . Чем меньше значение - тем больше значение N 0 . Например, если , то N 0 =999 .

We now give two equivalent definitions of the limit of the function : using the limit of the sequence and using the correspondence of small neighborhoods of the argument and the value of the function. The validity of one definition implies the validity of another. Let the function y=f(x) be defined , except maybe the point x=x 0 , which is the limit point of D(f) . At this point, the function may be undefined (undefined) or may have a break.

If the sequence converges to zero:

then it is called an infinitesimal sequence. It is also said that its common term is at an infinitesimal quantity. The sequences (84.3) and (84.4) are infinitesimal.

If we apply the formulation of the concept of limit to the case of an infinitely small sequence, i.e., to the case when the limit is equal to zero, then we arrive at the following definition of an infinitely small sequence (equivalent to the one given above): a sequence is called infinitesimal if for any given there is such a number N, that for all there will be an inequality

Let us formulate some useful theorems about infinitesimal sequences (and prove the first of them as an example).

Theorem 1. The sum of two or more infinitesimal sequences is an infinitesimal sequence.

We carry out the proof for the case of summation of two sequences. Let the sequences be infinitesimal. If is the sequence obtained by their addition, then it will also be infinitesimal. Indeed, let an arbitrary positive number e be given. Due to the fact that it is infinitely small, there is a number N such that it will be less than the number at . Similarly, for the second sequence, one can specify a (generally speaking, different) number such that for we have Now, if greater than the largest of the numbers , then simultaneously

But then, by the property "the modulus of the sum does not exceed the sum of the modules" (item 74, property 13), we find

which proves the required assertion: the infinitesimal sequence is read as “the larger of the two numbers N and .

Theorem 2. The product of a bounded sequence and a sequence converging to zero is a sequence converging to zero.

From this theorem, in particular, it follows that the product of a constant value by an infinitesimal, just as the product of several infinitesimals by each other, is an infinitesimal quantity. Indeed, a constant value is always a limited value. The same applies to the infinitesimal. Therefore, for example, the product of two infinitesimals can be interpreted as the product of an infinitesimal by a bounded one.

Theorem 3. The quotient of dividing a sequence that converges to zero by a sequence that has a non-zero limit is a sequence that converges to zero.

The following theorem allows the use of infinitesimals in the proofs of theorems on limits (Theorems 6-8).

Theorem 4. The common term of a sequence that has a limit can be represented as the sum of this limit and an infinitesimal quantity.

Proof. Let there be a sequence such that

From the definition of the limit follows:

for all satisfying the inequality Denote and then we get that for the indicated values ​​it will be

i.e., that there is an infinitesimal quantity. But

and this proves our theorem.

Verna and reverse

Theorem 5. If a common term of a sequence differs from some constant value by an infinitesimal value, then this constant is the limit of this sequence.

We now consider the rules for passing to the limit formulated in the following three theorems.

Theorem 6. The limit of the sum of two or more sequences that have a limit is equal to the sum of these limits:

Proof. Let there be sequences such that

Then, based on Theorem 4, we can write:

where are some infinitesimal sequences. Let's add the last two equalities:

The value as the sum of two constants a and b is constant, and as the sum of two infinitesimal sequences, according to Theorem 1, there is an infinitesimal sequence. From this and Theorem 5 we conclude that

and this was to be proved.

The proof we have now carried out can be easily generalized to the case of an algebraic sum of any number of given sequences.

Let the price of some asset at the current moment of time r be equal to S(T) . The exercise price of a call option on this asset with expiration time T is equal to K. Let us calculate the price of this option at time t. Divide the time interval [r, T] into n periods of the same length (T - t)/n. Calculation of the call option price is carried out within the framework of the n-period binomial option pricing model, and then its limit is found at n -> oo.
So, the option price in the n-period binomial model is determined by formula (3.12). According to the definition, jo tends to In [K/(S(t)dn))/ ln(m/d) as m i —» oo. According to the Moivre-Laplace integral formula
b&j0,n,p) - 1 -F (, b&j0,n,p") -
y/npq J \ l/np"q
where Ф(х) = ^ dt - normal distribution function.
Using the definition (3.16) of the numbers and ad, we obtain that as η -> oo
c \u003d S (r) Ф (гіі) - Ke-r ^-T4 (d2), (3.17)
where
\ii(S(t)/K) + (r + a2/2)(T - m)
d\
al/T - t
al/T - t
The found formula (3.17) for the call option price is called the Black-Scholes formula.
The proof of formula (3.17) uses the expansion of the exponent in the series
ex = 1 + x+^+.... (3.18)
Substituting and and d from formula (3.17) into equality (3.8), which determines the numbers р id, we get:
erAt - ate/Sh-
R
Expanding the exponentials into a series according to formula (3.18) and neglecting the terms that are small compared to At, we obtain
al / At + (g - a212) At al / At - (g - a212) At
P ~ t= 1 I ~ t=
2al/M 2al/M
If there is no market price uncertainty, then the asset price S satisfies the equation
AS = fiSAt, (2.1)
where At is small enough. As At -> 0 equation (2.1) becomes differential
S" = /J.S.
Its solution S(T) = S(0)emT determines the price S(T) of the asset at time T.
In practice, however, there is always uncertainty about the price of an asset. To describe the uncertainty, time functions are considered, which are random variables for each value of the argument. This property defines a random process.
A random process w(t) is called Wiener if r(0) = 0 and the random variables w(t\ + s) - w(t\) and w(t2 + s) - w(t2) have a normal distribution with zero expectation and with variance equal to s and are independent for any t\, t2, s forming non-overlapping intervals (ti,ti + s) and (t2,t2 + s).
The graph of the Wiener process can be obtained, for example, as follows. We fix some number h > 0 and define a family of random variables Wh(t) at times t = 0, h, 2h,.... Set Wh(0) = 0. Difference AWh = Wh((k+l)h) - Wh(kh) is a random variable and is given by the table: AWh -6 6 P 1/2 1/2 coins. Then the mathematical expectation of the random variable AWh is M(AI//1) = 0, and the variance D(AWh) = S2. The number d is set equal to Vh so that the variance ~D(AWh) is equal to h.
It turns out that the Wiener process w(t) is obtained from the family of random variables Wh(t) as h -> 0. The passage to the limit itself is rather difficult and is not considered here. Therefore, the graph of the family Wh (t) for small h is a good approximation of the Wiener process. For example, for a visual representation of the Wiener process on a segment, it suffices to take h = 0.01.
In the simplest case, when /x = 0, that is, the stock market does not grow and does not decrease on average, it is assumed that
AS = aS Aw,
where w(t) is a Wiener process and a > 0 is some positive number. The fact that asset price increments are proportional to price expresses the natural assumption that the uncertainty of the expression (S(t + At) - S(t))/S(t) does not depend on S. This means that the investor is equally unsure which you get a share of profit at an asset price of $20 and at an asset price of $100.
The asset price behavior model is generally determined by the equation
A S(t) = /j,S(t)At + aS(t)Aw, (2.2)
The coefficient a, which is a unit of uncertainty, is called volatility.
2.2.

More on the topic of Limit transition:

1. The transition to a market economy is associated with the transition to a system of modern management, the main object of which is the organization (enterprise), and within it - the worker, the worker.
2. Limiting value (limiting value of an economic indicator)

Quantum mechanics contains the classical as a limiting case. The question arises as to how this passage to the limit is carried out.

In quantum mechanics, an electron is described by a wave function that determines various values ​​of its coordinate; the only thing we know so far about this function is that it is a solution of some linear partial differential equation. In classical mechanics, however, an electron is considered as a material particle moving along a trajectory that is completely determined by the equations of motion. A relationship analogous in some sense to the relationship between quantum and classical mechanics takes place in electrodynamics between wave and geometric optics. In wave optics, electromagnetic waves are described by vectors of electric and magnetic fields that satisfy a certain system of linear differential equations (Maxwell's equations). In geometric optics, the propagation of light along certain trajectories - rays is considered.

Such an analogy allows us to conclude that the passage to the limit from quantum mechanics to classical mechanics occurs similarly to the transition from wave to geometric optics.

Let us recall how this last transition is carried out mathematically (see II, § 53). Let and be one of the field components in an electromagnetic wave. It can be represented as and - with real amplitude a and phase (the latter is called eikonal in geometric optics). The limiting case of geometric optics corresponds to small wavelengths, which is mathematically expressed by a large amount of change at small distances; this means, in particular, that the phase can be considered large in its absolute value.

Accordingly, we proceed from the assumption that the limiting case of classical mechanics corresponds in quantum mechanics to wave functions of the form , where a is a slowly changing function, and takes large values. As is known, in mechanics the trajectory of particles can be determined from the variational principle, according to which the so-called action 5 of a mechanical system must be minimal (principle of least action). In geometric optics, the path of the rays is determined by the so-called Fermat principle, according to which the "optical path length" of the beam, i.e., the difference between its phases at the end and at the beginning of the path, should be minimal.

Based on this analogy, we can assert that the phase of the wave function in the classical limiting case should be proportional to the mechanical action S of the physical system under consideration, i.e. it should be . The proportionality coefficient is called the Plant constant and is denoted by the letter . It has the dimension of action (because it is dimensionless) and is equal to

Thus, the wave function of an "almost classical" (or, as they say, semiclassical) physical system has the form

Planck's constant plays a fundamental role in all quantum phenomena. Its relative value (compared to other quantities of the same dimension) determines the “degree of quantumness” of this or that physical system. The transition from quantum to classical mechanics corresponds to a large phase and can be formally described as a transition to the limit (just as the transition from wave to geometric optics corresponds to the transition to the limit of zero wavelength,

We have clarified the limiting form of the wave function, but the question still remains of how it is related to classical motion along a trajectory. In the general case, the motion described by the wave function does not at all turn into motion along a certain trajectory. Its connection with classical motion lies in the fact that if at some initial moment the wave function, and with it the probability distribution of coordinates, is given, then in the future this distribution will “move” as it should be according to the laws of classical mechanics (for more details, see end of § 17).

In order to obtain motion along a certain trajectory, it is necessary to start from a wave function of a special form, noticeably different from zero only in a very small section of space (the so-called wave packet), the dimensions of this section can tend to zero along with d. Then it can be argued that in the semiclassical case the wave packet will move in space along the classical trajectory of the particle.

Finally, quantum mechanical operators in the limit must be reduced simply to multiplication by the corresponding physical quantity.

Some function f will tend to the number A as x tends to the point x0 when the difference f(x) - A is arbitrarily small. In other words, the expression |f(x) –A| becomes less than any preassigned fixed number h > 0, as the modulus of the argument increment |∆x| decreases.

Limit transition

Finding this number A from the function f is called passage to the limit. In the school course, the passage to the limit will occur in two main cases.

1. Passing to the limit with respect to ∆f/∆x when finding the derivative.

2. When determining the continuity of a function.

Function continuity

A function is called continuous at x0 if f(x) tends to f(x0) as x tends to x0. In this case: f(x) – A = f(x) – f(x0) = ∆f.
This means that |∆f| will be small for small |∆x|. In words, small changes in the argument correspond to small changes in the value of the function.

Functions that are found in a school mathematics course, for example, a linear function, a quadratic function, a power function, and others, are continuous at every point in the area on which they are defined. For these functions, the graphs are depicted as continuous curved lines.

This fact is the basis of the method of constructing a graph of a function "by points", which we usually use. But before using it, it is necessary to find out whether the function under consideration is really continuous. For simple cases, this can be done based on the definition of continuity that we gave above.

For example: we will prove that a linear function is continuous at every point of the real line y = k*x + b.

By definition, we need to show that |∆f| becomes less than any preassigned number h>0, for small |∆x|

|∆f| = |f(x0 +∆x) – f(x0)| = |(k*(x0+ ∆x) +b) – (k*x0+ b)| =|k|*|∆x|.

If we take |∆x| >h/|k| for k not equal to zero, then |∆f| will be less than any h>0, which was to be proved.

Limiting rules

When using the limit transition operation, one should be guided by the following rules.

1. If the function f is continuous at the point x0, then ∆f tends to zero as ∆x tends to zero.

2. If the function f has a derivative at the point x0, then ∆f/∆x tends to f’(x0) as ∆x tends to zero.

3. Let f(x) tend to A, g(x) tend to B as x tends to x0. Then:

f(x) + g(x) tends to A + B;