【精品文档】411关于数学应用数学教育师范专业有关的外文文献翻译成品：幂级数的展开及其应用（中英文双语对照）

此文档是毕业设计外文翻译成品（ 含英文原文+中文翻译），无需调整复杂的格式！下载之后直接可用，方便快捷！本文价格不贵,也就几十块钱！一辈子也就一次的事！外文标题：Power Series Expansion and Its Applications外文作者：Walter.Rudin文献出处: Function Theory in the Unit Ball of Cn,2013,45-51 (如觉得年份太老，可改为近2年，毕竟很多毕业生都这样做)英文774单词，中文1323汉字。（如果字数多了，可自行删减，大多数学校都是要求选取外文的一部分内容进行翻译的。）Power Series Expansion and Its ApplicationsIn the previous section, we discuss the convergence of power series, in its convergence region, the power series always converges to a function. For the simple power series, but also with itemized derivative, or quadrature methods, find this and function. This section will discuss another issue, for an arbitrary function, can be expanded in a power series, and launched into.Whether the power series as and function? The following discussion will address this issue.1 Maclaurin (Maclaurin) formulaPolynomial power series can be seen as an extension of reality, so consider the function can expand into power series, you can from the functionand polynomials start to solve this problem. To this end, to give here without proof the following formula.Taylor (Taylor) formula, if the function at in a neighborhood that until the derivative of order, then in the neighborhood of the following formula: (9-5-1)Among That for the Lagrangian remainder. That (9-5-1)-type formula for the Taylor.If so, get , (9-5-2)At this point, ().That (9-5-2) type formula for the Maclaurin.Formula shows that any function as long as until the derivative, can be equal to a polynomial and a remainder.We call the following power series (9-5-3)For the Maclaurin series.So, is it to for the Sum functions? If the order Maclaurin series (9-5-3) the first items and for, whichThen, the series (9-5-3) converges to the function the conditions.Noting Maclaurin formula (9-5-2) and the Maclaurin series (9-5-3) the relationship between the knownThus, whenThere,Vice versa. That if,Units must.This indicates that the Maclaurin series (9-5-3) to and function as the Maclaurin formula (9-5-2) of the remainder term (when).In this way, we get a function the power series expansion:. (9-5-4)It is the function the power series expression, if, the function of the power series expansion is unique. In fact, assuming the function f(x) can be expressed as power series, (9-5-5)Well, according to the convergence of power series can be itemized within the nature of derivation, and then make (power series apparently converges in the point), it is easy to get.Substituting them into (9-5-5) type, income and the Maclaurin expansion of (9-5-4) identical.In summary, if the function f(x) contains zero in a range of arbitrary order derivative, and in this range of Maclaurin formula in the remainder to zero as the limit (when n ,), then , the function f(x) can start forming as (9-5-4) type of power series.Power Series,Known as the Taylor series.Second, primary function of power series expansionMaclaurin formula using the function expanded in power series method, called the direct expansion method.Example 1 Test the functionexpanded in power series of .Solution because,Therefore,So we get the power series, (9-5-6)Obviously, (9-5-6)type convergence interval , As (9-5-6)whether type is Sum function, that is, whether it converges to , but also examine remainder . Because ()，且,Therefore,Noting the value of any set ,is a fixed constant, while the series (9-5-6) is absolutely convergent, so the general when the item when , , so when n , there,From thisThis indicates that the series (9-5-6) does converge to, therefore ().Such use of Maclaurin formula are expanded in power series method, although the procedure is clear, but operators are often too Cumbersome, so it is generally more convenient to use the following power series expansion method.Prior to this, we have been a function, and power series expansion, the use of these known expansion by power series of operations, we can achieve many functions of power series expansion. This demand function of power series expansion method is called indirect expansion.Example 2 Find the function,Department in the power series expansion.Solution because,And，（）Therefore, the power series can be itemized according to the rules of derivation can be,（）Third, the function power series expansion of the application exampleThe application of power series expansion is extensive, for example, can use it to set some numerical or other approximate calculation of integral value.Example 3 Using the expansion to estimatethe value of.Solution because Because of, (),So thereAvailable right end of the first n items of the series and as an approximation of . However, the convergence is very slow progression to get enough items to get more accurate estimates of value.