Infinite series is the study of ordered countable or the sum of infinitely numbered functionsastringencyThe theory is based on several term series, which has the difference between divergence and convergence.There is a sum only when the infinite series converges, and there is no sum for the divergent infinite series.
Approximating a function in an analytical form is generally to use a simpler form of function to approximate a more complex function. The simplest approach is throughadditionThat is, the degree of approximation is determined by addition, or the process of approximation is controlled, which is the starting point of the idea of infinite series.
Infinite series is the study of the sum of ordered countable infinite functionsconvergenceSex andlimitValue method, the theory is based onNumber seriesBased on, there are differences between divergence and convergence in the number of terms of series.When the infinite series converges, there is a unique sum;The divergent infinite series has no limit value, but there are other summation methods, such as Euler sum, Chesarro sumBorrell andwait.[1]
Arithmetic addition can sum finite numbers, but cannot sum infinite numbers. Some number sequences can be summed by infinite series method.The infinite series method can be used for summation, including: number series, function series (also includingpower series、fourier series ；Complex functionIntaylor series 、Loran series。[2]
history
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britainUniversity of Manchesteranduniversity of exeter The research team ofKerala SchoolIt has also been found that it can be used for calculationPiAnd use it to make the value of pi accurate to the 9th and 10th place after the decimal point, and then to the 17th place.The researchers said that a very persuasive indirect evidence is that in the 15th century, Indians once told their findings to math experts who visited Indiathe society of jesusmissionary."Infinite series" may end upNewtonOn my desk.
JosephThese findings were made when reading through Indian writing materials with ambiguous handwriting. The third edition of his best-selling book The Crest of the Peacock: the Non European Roots of Mathematics will publish this discovery, which was published byPrinceton University The publishing house is responsible for publishing.He said:“modern mathematics The origin of the ancient Indian literature is usually regarded as an achievement made by Europeans, but these discoveries in medieval India (from the 14th to the 16th century) were ignored or forgotten.At the end of the 17th century, Newton made brilliant achievements in his work.His contributions can not be obliterated, especially when it comes to calculusAlgorithmThis is especially true when.butkerala School scholars - especiallyMarderMadhava and NylaCanterThe name of Nilakantha should also not be forgotten. Their achievements are enough to matchNewtonThey are equal, because they discovered another important part of calculus, the infinite series. "[3]
Practical power series
JosephMeans:“kerala There are many reasons why the school's contributions have not been recognized by the world. One of the most important reasons is the indifference to scientific discoveries in the non European world. This practice is undoubtedly a continuation of European colonialism in the field of science.In addition, for the Kerala language in the Middle AgesMalayalamOutsiders can be said to know little about the forms of Indian local languages, but some of the most pioneering works, such as Yuktibhasa, use these languages.《Yuktibhasa》Most of the space in is devoted to describing the infinite series that have an important impact. "He pointed out: "We really can't imagine that the western society can abandon the 500 year old tradition and 'import' knowledge and works from India and the Islamic worldJesuitHaving visited this area, they have many opportunities to collect relevant information.More importantly, these Jesuits were proficient not only in mathematics, but also in the local language.
JosephSaid, "The reason why they did so was actually very motivated: Pope Gregory VIII set up a committee to modernize the Julian calendar in Rome. The members of this committee include German Jesuit, astronomer and mathematician Klavius, who has repeatedly asked for information on how people in other parts of the world build calendarskerala The school undoubtedly plays a leading role in this field. "He said: "Similarly, people's demand for more effective navigation methods has become stronger, including how to maintain the accuracy of time during the expedition. In addition, mathematicians who are committed to astronomical research can also win prizes with their own efforts.Therefore, the important Jesuit researchers in Europe began to travel all over the world to obtain relevant knowledge and information, and the mathematicians in Kerala School were undoubtedly masters in this field.Wangba S "
The above expression is calledInfinite series of constant terms（infinite series）, short forseries, recorded as
Where the
term
It is called the general term or general term of the series.[4]
Generally speaking, we have
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The infinite series has the following properties:
1. A necessary condition for the convergence of a series is that its general term is 0limit。
Certificate:
2. If there is an infinite series:
If there is another infinite series: multiply each term by oneconstant
, the sum is equal to
。I.e
3. Convergent series can be added or subtracted item by item. If there are two infinite series:
,
be
, which can be deduced from the addition and subtraction property of limit
4. The removal or addition or change of finite terms in a series does not affect its convergence,
For example:
and
The convergence and divergence of these two series are the same, but the limit values are not necessarily equal.
5. Partial sum sequence of convergence series
Subsequence of
It also converges (the inverse negative proposition also holds), and its sum is the sum of the original number;if
Convergence, then
Not necessarily convergent.
6. 3: If any finite infinite series are convergent, then they are arbitrarylinear combinationIt must also be convergent.Note that there is no similar conclusion for the series that are all divergent.
7. Inference of 5: If the series converges, then
Convergence, the corresponding new series (general term:
）Must converge (the inverse negative proposition also holds);If only
Convergence, the series may not converge.
power series
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If the series obtained from the absolute value of an arbitrary term series converges, the original series will also converge. If it diverges, the original series will not necessarily diverge. If it converges instead, it is called conditionally convergent.actually,Conditional convergenceThis series can converge to anyreal numberAnd can also radiate to infinity.[5]
Convergence radius and convergence interval
Each term of a series can also be a function. This series is called a functionNumber series。
Here we discuss a specific series of function terms, that is, the series composed of power functions in the following form, called power series.
It can also be written directly.The convergence and divergence of power series are characterized by the so-calledAbel's theorem: If the power series converges at the point x=k, then itsectionEvery point insideAbsolute convergence；On the contrary, if the power series diverges at the point x=k, then it diverges for all x that do not belong to.The above theorem makespower functionThe convergence domain of a power series can only be an open interval, which is called the convergence domain of a power series.Half of the length of the convergence interval is calledRadius of convergence。Application of ratio discrimination and root value discrimination for positive term serieslimitThe convergence radius of the power series can be found.
When the power series is absolutely convergent, and
The power series is divergent.Therefore, the convergence radius of power series is
。
⑵ If
, then for any x, the power series is absolutely convergent.Because at this time
, less than 1.At this time, we can consider that the convergence radius of power series is infinite.
⑶ If
If it is infinite, the power series will converge only at x=0, and if any non-zero value is taken, the series will be divergent, so it can be considered that the convergence radius of the power series is 0.
To form
Similarly, the power series oflimitForm, the same conclusion is obtained.After finding the convergence radius of the power series, the corresponding convergence can be obtainedsectionAnd convergence region.
Differential, integral and continuity of power series
Infinite series
For apower seriesIf its convergence radius is greater than 0, then in its convergence region, we get a certainDefine FieldsThe function of is the sum function of this power series. Naturally, this sum function should also be applicableCalculusTo study.
If the convergence radius r of is greater than 0, its sum function S (x) must be integrable on (- r, r), and the derivative function is
。Of and functionsDifferentiablesectionIs an open interval, because even if theEndpointThere may be definitions, and this theorem cannot guarantee the differentiability of sum function at the endpoint.
And functions also haveContinuity: If the convergence radius r of a power series is greater than 0, its sum function S (x) is continuous in its domain.For continuity, the theorem emphasizes on its domain, that is, including defined endpoints.Continuity means thatpower seriesFind the limit item by item.
Taylor series is a special form of Laurent series.
Divergent property
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First, we only consider the convergence and divergence of the series, that is, the existence problem, rather than how to findlimitProblems.The convergence and divergence of infinite series have the following basic properties:
Any change in the value of any finite term of a series does not affect theConvergence divergence。The reason is obviousseriesIf the changes made are finite, they cannot make this series become finite from tending to infinity, nor can they make this series become infinite from tending to infinity, or become a limit from having no limit at all.[6]
Discriminant method
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Positive Term Series and Its Convergence and Divergence
If every term of an infinite series is greater than or equal to 0, the series is called a positive term series.
Positive seriesThe main feature ofseries, you get oneMonotonous riseSeries.It is easy to judge the convergence and divergence of monotone ascending sequence:
The necessary and sufficient condition for the convergence of positive series is that the sequence of partial sums is bounded.
BoundednessThere are many ways to judge, from which we can get a series of convergence and divergence discrimination methods.
(1) For a positive term series, if after a finite term, all terms are less than or equal to the corresponding term of a known convergent series, then the positive term series will also converge.
(2) On the contrary, for a positive term series, if after a finite term, all terms are greater than or equal to the corresponding term of a known divergent series, then the positive term series must also be divergent.
If the term by term comparison is still troublesome, the following limit form can be used: for positive series and, if the ratio of their general terms tends to a finite value of non-zero, then the two series have the same convergence and divergence.
integral
For positive series, if there is a monotonic declinecontinuous function F (x), Yes
For positive term series, if starting from a certain term
, then the series converges; on the contrary, if there is a term satisfying
, the series is divergent.Similarly, this comparison can also be in the limit form: let the positive term series start from a certain term,
, the series is absolutely convergent.if
, the series is divergent.If it is not the above two, this method cannot be used for judgment.
Absolute convergence
In fact, the effective range of the convergence divergence discrimination method for positive series can be expanded, that is, it can also be used to judge that more series are convergent.This is obtained by introducing the concept of absolute convergence.
If we take the absolute value of each term of a series with any term, then we get a positive term series. If the positive term series is convergent, then the series with any term is said to be absolutely convergent.
The advantages of absolute convergence of such a class of arbitrary term series lie in:
If a series is absolutely convergent, then it must be convergent.
The absolutely convergent series not only has the characteristics that the discriminant method for the convergence and divergence of positive series can be applied, but also has the following properties:
If all positive terms of any term series are kept unchanged, and all negative terms are replaced with 0, then a positive term series is obtained;If all its negative terms change the sign, and the positive term is replaced with 0, then another positive term series is obtained, and then the necessary and sufficient condition for the absolute convergence of an arbitrary term series is obtained, which is positive term series and convergence.From this property, we can get a corollary, that is, if the series of any term is absolutely convergent, there will be.
AsAdditive commutative lawFor a positive term series, if you arbitrarily change the order of addition of its terms, it will not change its convergence and divergence. Similarly, for an absolutely convergent series, it also has this property.
Not only for the commutative law of addition, but also for the product of absolutely convergent series:
If two arbitrary series are absolutely convergent, then the product of their terms is also absolutely convergent, and the sum is the product of the sum of two arbitrary series.
Staggered series
Consider a special form of series, that is, the signs of two adjacent terms are opposite, calledStaggered series。Staggered series has a simple property:
If
Is a monotonedecreasing sequence , and take 0 aslimit, then the two staggered series constructed by changing the sign of the two adjacent terms of the sequence converge.
