Then the expression composed of this series.

It is called (constant term) infinite series, or (constant term) series for short, and it is written as follows, namely

The first term is called the general term of the series.

Find the sum of the antecedents of (constant term) series.

It is called the partial sum of series, and when taken in turn, they form a new series.

If the sum of the parts of a sequence has a limit, that is

When an infinite series converges, the limit is called the sum of the series, and it is recorded as

If there is no limit, it is called infinite series divergence.

Obviously, when a series converges, its partial sum is an approximation of the sum of the series and the difference between them.

It is called the remainder of series, and the error generated by replacing sum with approximation is the absolute value of this remainder, that is, the error is.

Property 1? If the series converges to the sum, the series also converges, and the sum is.

Conclusion: After each term of series is multiplied by a constant, its convergence will not change.

Nature 2? If the series converges to the sum, the series also converges, and its sum is.

Conclusion: The two convergent series can be added and subtracted item by item.

Nature 3? Removing, adding or changing the finite term in the series will not change the convergence of the series.

Nature 4? If the series converges, it is a series formed by arbitrarily adding brackets to the items of this series.

Or convergence, and unchanged.

Inference? If the series after parentheses diverges, the original series also diverges.

Property 5 (a necessary condition for series convergence)? If the series converges, its general term tends to zero, that is

Cauchy convergence principle? The necessary and sufficient condition for the convergence of series is that there are always positive integers for any given positive integer, so that there are positive integers at any time.

The necessary and sufficient condition for the convergence of theorem 1 positive series is that its partial sum series is bounded (a series with all positive or zero terms is called a positive series).

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Theorem 2 (Comparative Convergence Method)? Let sum be a positive series, the series will converge if it converges, and the series will diverge if it diverges.

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Inference? Let sum be a positive sequence. If the series converges and has a positive integer, the series converges. If the series diverges and holds, the series diverges.

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Theorem 3 (limit form of comparative convergence method)? Let sum be a positive sequence,

(1) If the sum series converges, the series converges;

(2) If or and the series diverge, the series diverges.

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Theorem 4 (ratio convergence method D 'Alembert discriminant method)? Set to positive series, if

Then when the time series converges and diverges, the time series may converge or diverge.

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Theorem 5 (root value convergence method Cauchy discriminant method)? Set to positive series, if

Then when the time series converges and diverges, the time series may converge or diverge.

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Theorem 6 (Limit Convergence Method)? Set it as a forward sequence,

(1) If, the series diverges.

(2) If and, the series converges.

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Theorem 7 (Leibniz Theorem)? If the interleaving sequence meets the condition:

( 1) ;

(2)

Then the series converges, and its sum is the absolute value of the remaining term.

(The term staggered series is positive and negative staggered)

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Absolute convergence and conditional convergence? If the positive series formed by the absolute values of the series items converges, it is called absolute convergence of the series; If the series converges and diverges, it is said that the series is conditionally convergent.

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Theorem 8? If the series absolutely converges, the series must converge.

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Theorem 9? The series formed by changing the position of the terms of the absolutely convergent series also converges and has the same sum as the original series (that is, the absolutely convergent series is commutative).

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Theorem 10 (multiplication of absolutely convergent series)? Let sum be absolutely convergent, and their sums are sum respectively, then their Cauchy products.

It is also absolutely convergent, and its sum is 0.

Given a function column defined on an interval.

Then the expression composed of this function column

It is called (function term) infinite series defined on the interval, or (function term) series for short.

In the convergence domain, the sum of series of function terms is a function, which is called the sum function of series of function terms, and is denoted as

A series of function terms whose terms are power functions is called power series, and its form is

Where the constant is called the power series coefficient.

Theorem 1 If the power series converges in time, then everything suitable for inequality makes the power series converge absolutely. On the other hand, if the series diverges in time, then everything suitable for inequality makes this power series diverge.

Inference? If the power series not only converges to one point or the whole number axis, then there must be a definite existence, so

If, the power series is absolutely convergent;

When, the power series diverges;

When, power series may converge or diverge.

Positive numbers are usually called the convergence radius of power series, and open intervals are called the convergence intervals of power series.

Theorem 2? if

Where is the coefficient of two adjacent terms of a power series, then the convergence radius of this power series.

Property 1? The sum function of power series is continuous in its convergence domain.

Nature 2? The sum function of power series is integrable in its convergence domain, and there is an integral formula item by item.

The power series obtained by item-by-item integration has the same convergence radius as the original series.

Nature 3? The sum function of power series is differentiable in its convergence interval, and there is a formula to derive the derivative item by item.

The power series obtained by item-by-item derivation has the same convergence radius as the original series.

Suppose that the function can be expanded into a power series in the domain of points, that is, there are

According to the properties of sum function, it is known that it has any derivative in sum.

From this, you can get

therefore

This shows that if the function has a power series expansion, then the coefficient of the power series is determined by the formula, that is, the power series must be

Expansion must be

Power series is called Taylor series of function at point, and expansion is called Taylor expansion of function at point.

Theorem? If a function has derivatives of various orders in the domain of a point, the necessary and sufficient condition for it to expand into Taylor series in this domain is that the limit of the remainder in Taylor formula is zero in this neighborhood, that is,

When, in the formula, take, get

Series is called maclaurin function series. If it can be expanded into a power series, there are

This formula is called the Ma Kraulin expansion of the function.

Commonly used power series expansion

From the two-sided integration to the formula, we can get

Derive both sides of the formula, that is.

Replace the formula with, and you can get

Replace the formula with, and you can get

By integrating the above formula from to, you can get

Binomial expansion

Let it be a periodic function with a period of, and it can develop a series called trigonometric series.

In ...

If all the integrals in exist, then the coefficients they determine are called Fourier coefficients of the function. Substituting these coefficients into the right side of the formula will get the trigonometric series.

Do Fourier series of functions.

When it is an odd function, it is an odd function and an even function, so

That is to say, odd function's Fourier series is a sine series containing only sine terms.

When it is an even function, it is an even function and a odd function, so

That is to say, the Fourier series of even functions is a cosine series containing only cosine terms.

Theorem (convergence theorem, Dirichlet sufficient condition)? Let be a periodic function with a period of, if it satisfies

(1) There are continuous or limited discontinuous points of the first kind in a period,

(2) There are at most a limited number of extreme points in a period,

Then the Fourier series converges, and

(1) When it is a continuous point, the series converges to;

(2) When it is a discontinuous point, the series converges to

Theorem? Let the period be a periodic function.