Taylor's formula focuses on those

Maclaurin formula): It can be deduced repeatedly by the law of Lobida.

f(x)=f(x0)+f'(x0)/ 1！ *(x-x0)+f''(x0)/2！ *(x-x0)^2+…+f^(n) (x0)/n！ (x-x0)^n+o((x-x0)^n)

Taylor mean value theorem (Taylor formula with Lagrange remainder): If the function f(x) has a first-order derivative of n+ 1 in the open interval (a, b) containing x, when the function is in this interval, it can be expanded into a function about the (x-x.) sum of a polynomial and a remainder;

f(x)=f(x .)+f'(x .)(x-x .)+f''(x .)/2! *(x-x .)^2,+f'''(x。 )/3! *(x-x .)^3+……+f(n)(x。 )/n！ *(x-x .)^n+Rn(x)

Where rn (x) = f (n+1) (ξ)/(n+1)! * (x-X.) (n+ 1), where ξ is between x and x, and the remainder is called Lagrange remainder.

(Note: f(n)(x). ) is f (x.), not f(n) and X. )

The conditions for using Taylor formula are: f(x)n order derivable. Where o ((x-x0) n) represents an infinitesimal of higher order than (x-x0) n. ..

The most typical application of Taylor formula is to find the approximate value of any function. Taylor formula can also find equivalent infinitesimal, prove inequality, find limit and so on.

Edit this paragraph to prove

We know that f(x)=f(x.)+f'(x.)(x-x.)+α (the finite increment theorem derived from Lagrange's mean value theorem is lim δ x → 0f (x.+δ x)-f (x.) = f' (x.) δ x, where the error α is in lim δ x.

p(x)=a0+a 1(x-x.)+a2(x-x.)^2+……+an(x-x.)^n

Approximate the function f(x) and write the specific expression of its error f(x)-P(x). Let the function P(x) satisfy p (x.) = f (x x.), p p' (x.) = f'' (x x.), p' (x.) = f'' (x.), ..., p (n) (x.) = f (n). Obviously, P(x.)=A0, so A0 = F (X.); P'(x.)=A 1，a 1 = f '(x .)； P''(x.)=2！ A2，A2=f''(x.)/2！ ……P(n)(x.)=n！ An，An=f(n)(x.)/n！ . So far, the coefficients of many terms have been calculated, which are: p (x) = f (x.)+f' (x.) (x-x.)+[f'' (x.)/2! ](x-x.)^2+……+[f(n)(x.)/n！ ](x-x.)^n.

Next, the specific expression of the error is needed. Let Rn(x)=f(x)-P(x), so rn (x) = f (x)-p (x) = 0. So it can be concluded that rn (x.) = rn' (x.) = rn'' (x.) = ... = rn (n) (x.) = 0. According to Cauchy mean value theorem, we can get rn (x)/(x-x.) (n+1) = (rn (x)-rn (x.)/(x-x.) (n+1)-0) = rn' (. Continue to use (rn' (ξ1)-rn' (X.))/((n+ 1) (ξ1-X.) n-0) = rn after continuous use of n+1times. Here ξ is between X. and X. But Rn (n+1) (x) = f (n+1) (x)-p (n+1) (x), because P(n)(x)=n! Ann, n! An is a constant, so P(n+ 1)(x)=0, so we get rn (n+1) (x) = f (n+1) (x). To sum up, the remainder rn (x) = f (n+1) (ξ)/(n+1)! ? (x-x.)^(n+ 1)。 Generally speaking, when expanding a function, it is for the need of calculation, so X often takes a fixed value, and then Rn(x) can also be written as Rn.

Maclaurin expansion

If the function f(x) has the first derivative up to n+ 1 in the open interval (a, b), when the function is in this interval, it can be expanded into the sum of a polynomial about x and a remainder;

f(x)=f(0)+f'(0)x+f''(0)/2！ ? x^2,+f'''(0)/3！ ? x^3+……+f(n)(0)/n！ ? x^n+Rn

Where rn = f (n+1) (θ x)/(n+1)! ? X (n+ 1), where 0

It is proved that if the polynomial p (x) = A0+A 1x+A2X 2+ is used to approximate the function f(x)...+ANX N and obtain the specific expression of its error, we can rewrite Taylor's formula into a simpler form, that is, a special form when x = 0:

f(x)=f(0)+f'(0)x+f''(0)/2！ ? x^2,+f'''(0)/3！ ? x^3+……+f(n)(0)/n！ ? x^n+f(n+ 1)(ξ)/(n+ 1)！ ? x^(n+ 1)

Since ξ is between 0 and x, it can be written as θx, 0.

Application of maclaurin expansion

：

1, expand trigonometric function y=sinx, y=cosx.

Solution: According to the derivative f (x) = sinx, f' (x) = cosx, f'' (x) =-sinx, f'' (x) =-cosx, f (4) (x) = sinx. ...

Thus, the periodic law is obtained. Calculate f(0)=0, f' (0) = 1, f'' (x) = 0, f'' (0) =- 1, and f (4) = 0. ...

Finally available: sinx = x-x 3/3! +x^5/5！ -x^7/7！ +x^9/9！ -(This is written in the form of infinite series. )

Similarly, y=cosx can also be expanded.

2. Calculate the approximate value e = lim x→∞ (1+1/x) x.

Solution: apply maclaurin expansion to exponential function y = e x and discard the remainder:

e^x≈ 1+x+x^2/2！ +x^3/3！ +……+x^n/n！

When x= 1, e≈ 1+ 1+ 1/2! + 1/3! +……+ 1/n！

If n= 10, the approximate value e≈2.7 1828 18 can be calculated.

3. Euler formula: e ix = cosx+isinx (I is the root of-1, that is, the imaginary unit).

Proof: This formula writes complex numbers as power exponents, which is actually proved by maclaurin expansion or McLaughlin series. I won't write the process in detail, but let's talk about the idea first: first expand the exponential function e z, and then write z in each term as ix. Because of the power periodicity of I, the term containing soil I in the coefficient can be written together by multiplication and division and distribution, and the remaining terms are also written together, which is exactly the expansion of cosx, sinx SINX. Then multiply sinx by the suggested I, and the Euler formula can be derived. You can prove it yourself if you are interested.

Edit the Taylor expansion principle in this paragraph.

The discovery of e began with differentiation. When h gradually approaches zero, the calculated value is infinitely close to a certain value of 2.7 1828 ..., which was first discovered by the famous Swiss mathematician e Euler. He named the irrational number after himself, prefixed with lowercase e.

Calculate the derivative of logarithmic function, and get that when a=e, the derivative of is 0, so the logarithm based on e is more reasonable, which is called natural logarithm.

If the exponential function ex is Taylor expansion, then

Substitute x= 1 into the above formula.

This series converges quickly, and the value of E is approximately 40 decimal places.

When the exponential function ex is extended to the complex number z=x+yi, it is determined by the following formula.

Through this series of calculations, we can get

From this, Demol's theorem, the formula of sum and difference angles of trigonometric functions and so on can be easily deduced. For example, z 1 = x 1+y 1i, z2 = x2+y2i,

On the other hand,

So,

We can not only prove that e is an irrational number, but also a transcendental number, that is, it is not the root of any integer coefficient polynomial. This result was obtained by Hermite in 1873.

A) differences.

Consider a discrete function (sequence) R, whose value u(n) at n is denoted as un, and we usually write this function as OR (un). The difference of the sequence U is still a sequence, and its value in n is defined as

In the future, let's remember Jane

(Example): The difference sequence of sequence 1, 4, 8, 7, 6, -2, ... is 3, 4,-1, -8. ...

Note: We say "series" is "a function defined at discrete points". This statement sucks in high school, but it is appropriate here because it has a completely parallel analogy with continuous functions.

The nature of difference operator

(a) [collectively referred to as linear]

(ii) (Constant) [Basic Theorem of Difference Equation]

㈢

Where (n(k) is called a permutation sequence.

(4) called natural geometric series.

(iv) The difference sequence (i.e. "derivative function") of the' general exponential sequence (geometric sequence) rn is rn(r- 1).

(2). Sum integral

Give a series of numbers (un). The problem of summation is to calculate summation. How to calculate? We got the following important results:

Theorem 1 (basic theorem of difference and division) If we can find a sequence (vn), then

Sum and fraction also have linear properties:

A) differences

Given a function f, if the limit of Newton's quotient (or difference quotient) exists, then we call this limit value f the derivative of point x0 and write it as f'(x0) or Df(x), that is.

If the derivative of f exists at every point in the defined area, it is called a differentiable function. We call it the derivative function of f, not the differential operator.

Properties of differential operators:

(a) [collectively referred to as linear]

(ii) (Constant) [Basic Theorem of Difference Equation]

(3) Dxn=nxn- 1

Dex = ex

(iv)' The derivative function of the general exponential sequence ax is

(b) integration.

Let f be a function defined on [a, b], and the problem of integration is to calculate the shadow area. Our method is divided into [a, b]:

; Secondly, take a sample point [xi- 1, xi] for each small segment; Then find the approximate sum; Finally, take the limit (let the length of each segment approach 0).

If this limit exists, we will remember that the geometric meaning is the shadow area.

(In fact, continuity is also "almost" a necessary condition for the existence of integral. )

Integral operators also have linear properties:

Theorem 2 If F is a continuous function, it exists. (In fact, continuity is also a necessary condition for the existence of an integral. )

Theorem 3 (Basic Theorem of Calculus) Let f be a continuous function defined in the closed interval [a, b], and we want to get the integral. If we can find another function g that makes g'=f, then

Note: (1) (2) Although the two formulas are analogies, there is one difference, that is, be careful about the upper limit of the sum!

Theorem 1 and Theorem 3 above basically talk about difference and division, and differential and integral are two reciprocal operations, just as addition, subtraction, multiplication and division are reciprocal operations.

As we all know, the operation of difference and differential is much simpler than the operation of sum, fraction and integral. Theorem 1 and Theorem 3 above tell us that to calculate the sum of (un) and the integral of fraction and f, we only need to find another (vn) and g to satisfy G'= F (this is a problem of difference and differentiation), and then we can get the answer by substituting vn and g into the upper and lower limits. In other words, we can use something simpler.

A) Taylor expansion formula

There are discrete and continuous analogies. It is a special case of the important idea of approximation in mathematics. The idea of approximation is this: given a function F, we should study its behavior, but F itself may be very complicated and difficult to handle, so we try to find a simpler function G to make it "close" to F, and then we use G instead of F, which is to simplify the complexity.

Two questions: How to choose simple function and approximate scale?

(1) In the case of continuous world, the approximation idea of Taylor expansion is to select polynomial function as simple function and local tangency as approximation scale. More specifically, given a function f that can be differentiated to order n, we need to find a polynomial function g of order n so that it is "tangent" to f at point x0, that is, the answer is

This formula is called the n-order Taylor expansion of f at point x0.

G is very close to F near x0, so we use G to replace F locally, so we can use G to find some local qualitative behaviors of F, so Taylor expansion is only a local approximation. When F is a good enough function, that is, the so-called analytic function, F can be expanded into Taylor series, which is equal to F itself.

It is worth noting that in the special case of the first-order Taylor expansion, the graph of g(x)=f(x0)+f'(x0)(x-x0) is just a straight line that passes through the graph of points (x0, f(x0)) and is tangent to F. Therefore, the significance of the first-order Taylor expansion of f at point x0 is that we have used the point (x0).

Taylor expansion can help us do many things, such as judging the maximum and minimum values of functions, finding the approximate value of integrals, and making function tables (such as trigonometric function tables and logarithmic tables). In fact, we can "consistently" calculus with approximate ideas.

Many times, we notice that we choose polynomial function as simple approximation function for a simple reason: among many elementary functions, such as trigonometric function, exponential function, logarithmic function, polynomial function and so on. From the arithmetic point of view, polynomial function is the simplest, because to calculate the value of polynomial function, it only involves four operations of addition, subtraction, multiplication and division, and other functions are not so simple.

Of course, from other analysis angles, in some cases, there are other simple functions that are more useful and important. For example, trigonometric polynomials, combined with some approximation scales, give us Fourier series expansion, which plays an important role in applied mathematics. (In fact, Fourier series expansion is an approximation scale with the smallest variance, which often appears in higher mathematics and is also applied in statistics. )

Note: Take the special case of x0=0 as an example. At this time, Taylor expansion is also called Ma Kraulin expansion. However, as long as we can expand a special case and want a general Taylor expansion, it is good to do translation (or variable substitution). Therefore, Taylor expansion can only be done at the point of x=0 from the beginning.

(2) For discrete cases, Taylor expansion is:

Given a sequence, we need to find a polynomial sequence with degree n (gt) so that gt and ft have N-order "difference approximation" when t=0. The so-called zero-order n-order difference approximation refers to:

The answer is that this formula is Maclaurin formula in discrete case.

B) analogy between partial integral formula and Abel partial sum formula

(a) partial integral formula:

Let u (x) and v (x) be continuous on [a, b], then

(2) Abel partial sum formula:

Let (UN) and (v) be two series, so Sn = U 1+...+UN, then

The above two formulas are Leibniz's derivative formula D(uv)=(Du)v+u(Dv) and Leibniz's difference formula respectively. Note that one of the two Leibniz formulas is very symmetrical, while the other is asymmetrical.

(d) compound interest and continuous compound interest (this is also an analogy between discrete and continuous respectively)

(1) The problem of compound interest is as follows: there is a principal y0, with an annual interest rate of R, and compound interest once a year. Ask the principal and interest after N years and yn= Obviously, this series satisfies the difference equation yn+ 1=yn( 1+r).

According to (2) of (c), we know that yn=y0( 1+r)n is a compound interest formula.

(2) If compound interest is considered m times a year, the sum of principal and interest after t years is

Order, you get the concept of continuous compound interest, and the sum of principal and interest at this time is y(t)= yoert.

In other words, the principal and interest of time t and y(t)= yoert are the solutions of the differential equation y'=ry.

As can be seen from the above, the discrete compound interest problem is described by difference equation, while the continuous compound interest problem is described by differential equation. For linear difference equations and differential equations with constant coefficients, the whole point of solving equations is the superposition principle, so the solving methods are completely parallel.

(e) Fubini's theorem of multiple sums and points and Fubini's theorem of multiple integrals (also an analogy between discrete and continuous).

(1) Fubini's double sum theorem: given a series of numbers (ars) with double exponents, we want to sum r= 1 to m and S = 1 to n (ars), then this sum can be obtained as follows: light sum r, and then sum s (and vice versa). In other words, we have.

(2) Fubini's multiple integral theorem: Let f(x, y) be an integrable function defined on, then

Of course, several variables are the same.

The concept of leberg integral

(1) Discrete case: Given a series (an), the sum should be estimated. Leberg's idea is that no matter what the order of the data indicators of this pile is, we only divide it into piles according to the size of the values, and then multiply a value from each pile by the number of the piles to get the overall sum.

(2) Continuity: Given a function f, we need to define the area enclosed by the curve y=f(x) and the X axis from A to B. 。

Leberg's idea is to divide the shadow domain of F:

X whose function value is between yi- 1 and yi is set together to make it 0, thus [a, b] is divided into sampling points and approximately summed.

If the limit of the above approximate sum exists, it is called Lebesgue integral of f on [a, b].

remainder

Taylor formula f(x)=f(a)+f'(a)(x-a)/ 1 remainder! + f''(a)(x-a)^2/2！ + …… + f(n)(a)(x-a)^n/n！ +Rn(x)[ where f(n) is the nth derivative of f]

Taylor remainder can be written in the following different forms:

1. Peano (piano) remainder:

Rn(x) = o((x-a)^n)

4. Remaining projects of 4.Schlomilch-Roche:

rn(x)= f(n+ 1)(a+θ(x-a))( 1-θ)^(n+ 1-p)(x-a)^(n+ 1)/(n！ p)

[f(n+ 1) is the n+ 1 derivative of f, θ∈(0, 1)]

3. Lagrange remainder;

rn(x)= f(n+ 1)(a+θ(x-a))(x-a)^(n+ 1)/(n+ 1)！

[f(n+ 1) is the n+ 1 derivative of f, θ∈(0, 1)]

4. Cauchy remainder:

rn(x)= f(n+ 1)(a+θ(x-a))( 1-θ)^n(x-a)^(n+ 1)/n！

[f(n+ 1) is the n+ 1 derivative of f, θ∈(0, 1)]

5. Integer remainder:

Rn(X)=[f(n+ 1)(t)(X-t)n an integer from a to x ]/n!

[f(n+ 1) is the n+ 1 derivative of f]

Edit Taylor's profile

Brook Taylor, an English mathematician,1one of the most outstanding representatives of the British Newton School in the early 8th century, was born in Edmonton, Oxfordshire, Delsey, England on August/865 1685. 170 1 year, Taylor entered St. John's College of Cambridge University. After 1709, he moved to London and obtained a bachelor's degree in law. 17 12 was elected as a member of the royal society. In the same year, he joined the committee urging Newton and Leibniz to debate the priority of inventing calculus. Received a doctorate in law two years later. Since 17 14, he has served as the first secretary of the royal society, and 17 18 resigned on health grounds. 17 17 He solved the numerical equation with Taylor theorem. Finally, he died in London on February 29th, 173 1.

Due to work and health reasons, Taylor visited France many times and corresponded with French mathematician Montemor many times to discuss the problems of series and probability theory. 1708, 23-year-old Taylor got the solution of "vibration center problem", which attracted people's attention. In this work, he used Newton's instantaneous mark. From 17 14 to 17 19, Taylor was in the period of mathematical Newton.

main work

His two books, Positive and Negative Increment Method and Linear Perspective Method, were published in 17 15, and the second editions were published in 17 17 and19 respectively. From 17 12 to 1724, he published 13 articles in the Journal of Philosophy, some of which were correspondence and comments. The article also contains the experimental records of capillary phenomenon, magnetism and thermometer.

In his later years, Taylor turned to writing about religion and philosophy. His third book, Philosophical Meditation, was published by his grandson W. Yang on 1793 after his death.

Taylor is famous for the theorem that functions are expanded into infinite series in calculus. This theorem can be roughly described as follows: the value of a function in the neighborhood of a point can be expressed by an infinite series composed of the value of the function at that point and the derivative values of each order. However, for half a century, mathematicians have not realized the great value of Taylor's theorem. This huge value was later discovered by Lagrange, who described this theorem as the basic theorem of calculus. The strict proof of Taylor theorem was given by Cauchy a century after the birth of the theorem.

Taylor theorem initiated the finite difference theory, so that any univariate function can be expanded into a power series; Meanwhile, Taylor became the founder of finite difference theory. Taylor also discussed the application of calculus in a series of physical problems, among which the result of transverse vibration of strings is particularly important. He deduced the basic frequency formula by solving the equation, which initiated the study of string vibration. In addition, this book also includes his other creative work in mathematics, such as discussing singular solutions of ordinary differential equations and studying curvature problems.

17 15 published another famous book, the theory of linear perspective, and even the second edition of the principle of linear perspective (17 19). He developed his linear perspective system in a very strict form, among which the most outstanding contribution was to put forward and use the concept of "vanishing point", which had a certain influence on the development of photogrammetry cartography. In addition, there is a philosophical legacy published in 1793.