Series refers toseriesA function whose terms are sequentially connected by a plus sign.Typical series include positive series, staggered series, power series, Fourier series, etc.
Series theory isAnalyticsA branch of;It is connected with another branchCalculusTogether, they appear in the rest of the branches as basic knowledge and tools.Both of them use limit as the basic tool to study the object of analysis, that is, the dependency relationship between variables - function, from the discrete and continuous aspects.
Chinese name
series
Foreign name
series
Full name
Number series
Field
mathematics
Meaning
It is an important tool for studying functions
Typical series
Positive series, staggered series, power series, etc
,... functions connected by the plus sign in turn areNumber seriesShort name of.For example:
, abbreviated as
，
It is called the general term of the series
It is called the partial sum of series.If when
When,seriesSn Yeslimit, the limit is S, then the series converges, and
Is the sum of them, recorded as
；Otherwise, the series is divergent.
Series is an important tool for studying functions, which plays an important role in both theory and practical application. This is because on the one hand, series can be used to express many common nonElementary functionThe solution of differential equation is usually expressed by series;On the other hand, the function can be expressed as a series, so that the function can be studied with the help of series, for example, usingpower seriesResearchNon elementary function, and approximate calculation.
Positive series
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The positive series represents the simplest case of convergence.In this case, the partial sum of the series sm=u1+u2+...+um monotonically increases with m, which is equivalent to the general term un ≥ 0 of the series (therefore, it is sometimes called a non negative term series).So the convergence of series (∑ un) is equivalent to the partial sum (sm)Bounded。The smaller the term, the more the partial sum tends to be bounded, so the positive term series hasComparative discrimination：
。
Similarly, the ratio of each term is smaller than that of the preceding term, and the partial sum also increases less and tends to be bounded. Therefore, the positive term series has ratio discrimination.In fact, it all depends on determining the magnitude of un.
Staggered series
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In addition to the positive term series, if a series has no positive term, or only a finite number of positive terms, or only a finite number of negative terms, its convergence problem can be reduced to the convergence problem of a positive term series, so only the case where a series has both infinite positive terms and infinite negative terms needs to be considered.In this series, the simplest structure isSignStaggered series:
。There areLeibnizTheorem: If the absolute value of the term of an alternating series monotonically tends to zero, the series converges.
Obviously, a staggered series can be regarded as the difference between two positive series in form.
Similarly, each series can be formally regarded as the difference between two positive series (i.e., the "positive part" and the "negative part" of the series):
However, such decomposition is meaningful only when the series decomposed are convergent, which leads people to consider whether the positive series obtained after a series is taken absolute value item by item is convergent.
power series
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An important class of function series is in the form of
The series of is calledpower series。Its structure is simple, and the convergence domain isbCentered interval (not necessarily includingEndpoint）, and similar within a certain rangepolynomialAnd can be carried out item by item within the convergence intervaldifferentialAnd item by item integration.For example, power series
The convergence interval of is (- 1/2, 1/2), the convergence interval of power series is (1, 3), and the convergence interval of power series is
SeriallyconvergenceThe problem is the basic problem of series theory.From the convergence concept of series, the convergence and divergence of series is defined by the convergence and divergence of its partial sum sequence Sm.So convergent from sequenceCauchyCriterion The Cauchy criterion of series convergence is obtained: ∑ un convergence<=>Any given positive number ε, there must be a natural number N. When n>N, for all natural numbers p, there is | u [n+1]+u [n+2]+...+u [n+p] |<ε, that is, the absolute value of any sum of sections sufficiently close to the back can be arbitrarily small.
Series convergence
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If each un ≥ 0 (or un ≤ 0), ∑ un is said to be a series of positive (or negative) terms,Positive seriesSeries with negative terms are collectively called series with the same sign.The necessary and sufficient condition for the convergence of positive series is that its partial sum sequence Sm hasupper bound, e.g. ∑ 1/n!Convergence, because: Sm=1+1/2+1/3!+···+1/m!<1+1+1/2+1/2 ²+···+1/2 ^ (m-1)<3 (2 ^ 3 means the 3rd power of 2).
The series with infinite multinomials being positive and infinite multinomials being negative are called sign changing series, among which the simplest is the series in the form of ∑ [(- 1) ^ (n-1)] * un (un>0), called staggered series.The basic method to judge the convergence of such series isLeibnizJudgment: if un ≥ un+1, it is true for every n ∈ N, and when n →∞, lim un=0, thenStaggered seriesConvergence.For example, ∑ [(- 1) ^ (n-1)] * (1/n) converges.For general sign changing series, if there is ∑ | un | convergence, it is called sign changing seriesAbsolute convergence。If only ∑ un converges, but ∑ | un | diverges, it is called sign changing seriesConditional convergence。For example, ∑ [(- 1) ^ (n-1)] * (1/n ^ 2) is absolutely convergent, while ∑ [(- 1) ^ (n-1)] * (1/n) is only conditionally convergent.
If each term of the series depends on the variable xsectionIf I changes internally, that is, un=un (x), x ∈ I, then ∑ un (x) is called a series of function terms, referred to as a series of function terms.If x=x0 makes the number term series ∑ un (x0) converge, it is called x0 as the convergence pointconvergenceThe set of points is called the convergence region. If the series ∑ un (x) converges for every x ∈ I, I is called the convergence region.Obviously, the function series defines a function in its convergence domain, called sum function S (x), that is, if S (x)=∑ un (x) meets stronger conditions, Sm (x) is in the convergence domainuniform convergenceOn S (x)[1]。
Absolute convergence
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brief introduction
A convergent series is said to be absolutely convergent if it still converges after taking the absolute value item by item;Otherwise, it is said to be conditionally convergent.
A simple comparison series shows that as long as ∑ | un | converges, it is enough to ensure the convergence of the series;Therefore, the decomposition formula (not only shows that the convergence of ∑ | un | implies the convergence of the original number ∑ un, but also expresses the original number as the difference between two convergent positive term series. It is easy to see that the absolute convergence series, like the positive term series, is very similar to the finite sum, which can be arbitrarily changed to sum the order of terms, and can be multiplied infinitely.
But the series of conditional convergence, that is, the series of convergence but not absolute convergence, can never do this.At this time, the right side of the formula becomes the difference between two divergent (to+∞) positive term series whose terms tend to zeroRiemannTheorem.
Riemann theorem
A conditionally convergent series can converge to an arbitrarily specified number in advance after its terms are properly arranged;It can also diverge to+∞ or - ∞;It can also be without any sum.
Uniform convergence is the most important form of the combination of convergence and function continuity.
