§11.5 function expands into power series

First, Taylor series

If there is a derivative of any order in place, we put the series

(1)

Called the Taylor series at the point where the function is.

The part of its preceding paragraph and its use is recorded, and

Over here:

The Taylor median theorem, which is described in the previous book, has

Of course, here is the Lagrangian residual item, and

。

by a

。

So, at that time, the Taylor series of functions

is another exact expression of it. That

At this point, we call the function to expand the Chentelle series at the place.

In particular, at that time,

At this point, we call the function can be expanded into Maclaurin series.

The function is expanded Chentelle series, which can be replaced by variable, and normalized to the maclaurin of function at the place. Therefore, we focus on the Maclaurin expansion of functions.

The Maclaurin expansion of the "propositional" function is unique.

Proof: A power series that can be expanded into a neighboring field

According to the power series in the convergence interval can be derivative, there are

Put in the formula, have

Thus

Thus, the function at the place of the power series expansion of its form is

This is the Maclaurin expansion of the function.

This shows that the function at the place of the Power series expansion form only Maclaurin expansion of this form.

Function expansion into power series

1. Direct Expansion method

The function can be expanded into Maclaurin series according to the following steps

œ the derivative of the function and the function value

If a derivative of a function does not exist, the function cannot be expanded;

Write Maclaurin series

And the convergent radius is obtained.

Z inspected the time, Lagrange

At that time, whether toward zero.

If, then the second step writes out the series is the function Maclaurin expansion type;

If, then the function cannot expand into Maclaurin series.

"Example 1" expands the function into Maclaurin series.

Solution:

So I got Maclaurin series.

and

So

For arbitrary, there

Here is a finite number with nothing to consider auxiliary power series

Convergence of the scattered nature. The ratio method has

Therefore the auxiliary series converges, thus the general term tends to be 0, namely

Therefore,

"Example 2" expands the function at the place into a power series.

Solution:

So he got a power series.

It is easy to find out that its convergent radius is

On the arbitrary, there

As we know from the example,

So, we get an expanded

2. Indirect expansion method

Using some known function expansion and the Operation property of power series (such as: addition and subtraction, derivation by item, product by item), the given function is expanded.

"Example 3" expands the function into a power series.

Solution: to the expanded type

On both sides of the derivative, get

"Example 4" expands the function into a power series.

Solution:

and

From the top to the point of integration

At that time, the staggered series

Convergence.

So

Next, we introduce the very important Newton two-item expansion

"Example 5" expands the function into a power series, which is any real number.

Solution:

And got a power series.

Therefore, the power series is convergent to any real number.

Here, we prove that the sum of the convergence of the power series is the function.

The sum function of the above power series is set, that is,

Both sides multiply by factor, have

That

Introducing Auxiliary functions

So, inside, we have an unfolded

Notes

The divergence of ¶ at the interval endpoint depends on the value of the real number, which we do not introduce further.

• If the generalized combination mark is introduced, Newton's two expansion type can be denoted into

Finally, we give an example of the form of a power series that expands the function.

"Example 6" expands the function into a power series.

Solution: As a variable substitution, then, there are

and

So

The application of §11.6 function in power series expansion

First, approximate calculation

The approximate calculation can be done by using the expansion of Function power series.

1. Some terms in approximate calculation

œ Error does not exceed

Set to a fine value, and an approximate value, the absolute error between and.

Error not exceeding () means:

The difference between the approximate value and the fine value, the bit after the decimal point is exactly the same, only the number of digits after the decimal point is not more than one unit.

For example:

Sometimes, the error does not exceed the said: accurate to the decimal place.

Truncation error (or method error)

Taylor Polynomial of function

To approximate substitution, the truncation error of the numerical calculation method is

¸ Rounding Error

Computer for numerical calculation, because the computer's word length is limited, the original data on the computer will produce errors, using these approximate data to calculate, but also may cause new errors, this error is called rounding error.

For example, using a 3.14159 approximation to replace p, the resulting error

D =p-3.14159 = 0.0000026L

is rounding error.

2, Radical calculation

"Example 1" calculates an approximate value (accurate to decimal four digits).

To find the approximate value of the radical, we should select the expansion of a function power series, and choose Newton's two expansion type.

To take advantage of this, you need to have a representation of the form, usually when smaller, the calculation effect will be better.

Here, desirable,.

Solution: Using two-item expansion, there

If we intercept the first four items for calculation, then

@

As the coefficients are monotonically decreasing, the truncation error can be estimated as follows

@

Indicate:

Ê expressions can also be selected in other forms, such as

；

E In the Limit theory study of series, we have been in the shape of the series

, it is monotonically descending, lower bound, and

Using this iterative formula, write MATLAB program GS1101.M, run this program, it is more easy to get the high accuracy approximate value.

3, the calculation of the logarithm

"Example 2" calculates the approximate value (accurate to the 4th digit after the decimal).

Solution: We already have an expansion

And

It is theoretically feasible to calculate the approximate value by using the series of these numbers. The part and the truncation error of the

In order to achieve precision, the number of items required should be met, namely

, that is, 10000 items should be taken, which is too great.

Run MATLAB program GS1102.M, take the first 10,000 items (n=10000) for approximate calculation, you can get the following table. The comparison between the number of items and the approximate values and the calculation speed are also observed carefully.

Number of items intercepted

LN2 approximate value

9900

0.6930971330

9991

0.6931972231

9992

0.6930971430

9993

0.6931972131

9994

0.6930971530

9995

0.6931972031

9996

0.6930971631

9997

0.6931971931

9998

0.6930971731

9999

0.6931971831

10000

0.6930971831

From the operation of the above procedures and results, there are several feelings

The ê part and the number of items obtained were too large to reach 10,000;

e its approximate value is only three digits after the decimal point is accurate;

The number of ì items increased by dozens of, and no significant digits were provided;

Í it takes too much time to calculate.

This forces us to look for more efficient methods of calculating LN2.

Expands the type

In the exchange, you have to

Two-type subtraction to obtain an expansion without even power

order, to solve. In lieu

Then, the program GS1103.M of the program in MATLAB, which can obtain the number of items and the approximate value of the algorithm, is as follows

Number of items intercepted

LN2 approximate value

4

0.69313475733229

5

0.69314604739083

6

0.69314707375979

7

0.69314717025601

8

0.69314717954824

9

0.69314718045924

10

0.69314718054981

11

0.69314718055892

12

0.69314718055984

13

0.69314718055993

14

0.69314718055994

15

0.69314718055994

16

0.69314718055995

17

0.69314718055995

18

0.69314718055995

It can be found from the table that the calculation speed is greatly improved, and the precision of approximation is improved remarkably, which is usually called the accelerating technique of power series convergence.

4, the calculation of P

In the elementary school mathematics study, we have contacted the Pi P, but has never actually done to its computation. Now is the time for us to have this ambition.

by an expanded

Points on both sides, there are

Order, then, then there

This formula can be used to calculate, the effect (speed and precision) is also good, just need the value. With the help of the triangular formula, it is necessary to construct an expression that does not need to be evaluated.

Order, there

According to the formula, write MATLAB program GS1104.M, run it can obtain the following results.

Number of items intercepted

P Approximate value

10

3.141592579606

11

3.141592670451

12

3.141592649717

13

3.141592654485

14

3.141592653382

15

3.141592653638

16

3.141592653578

17

3.141592653592

18

3.141592653589

19

3.141592653590

20

3.141592653590

5, approximate calculation of definite integral

"Example 3" calculating definite integral

The approximate value, accurate to 0.0001.

Solution: Because, the given integral is not a generalized integral, just define the function at the place of the value of 1, then it is continuous on the.

Expand Integrand, have

To point to points on the interval

Because the fourth

So the first three and the approximate value of the integral are desirable

For the above series expansion, we have written MATLAB program GS1105.M, run this program, can give the interception series any item, this definite integral contains more bits valid value approximate value.

Number of items intercepted

Approximate value of definite integral

1

1.00000000000000

2

0.94444444444444

3

0.94611111111111

4

0.94608276643991

5

0.94608307263235

6

0.94608307035488

7

0.94608307036723

8

0.94608307036718

9

0.94608307036718

10

0.94608307036718

Two, Euler formula

The series of plural items is provided with

(1)

Which is either a real constant or a real function. If the series of the real part

(2)

Convergent suitable which, and the series formed by the imaginary part

(3)

Convergent suitable which, it is said that the series (1) converges and its sum is.

If the series (1) of the modules of the series

(4)

Convergent, because

Then the series (2), (3) absolute convergence, so that the series (1) Convergence, then said series (1) Absolute convergence.

Examine the series of plural items

(5)

The series formed by its modulus

Absolute convergence. Therefore, the series (5) is absolutely convergent on the whole complex plane.

On the axis (), it represents an exponential function, which we use to define the exponential function of complex variables on the entire complex plane. then defined as

(6)

At that time, for pure imaginary numbers, (6) the type became