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1、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. T
2、his 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 rea
3、lity, 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 t
4、he 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 poly
5、nomial 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
6、 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
7、)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 (p
8、ower 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 remain
9、der 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
10、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,No
11、ting 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,
12、 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 a
13、chieve 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 deriva
14、tion 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 b
15、ecause 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.此外文文献选自于：Walter.Rudin.数学分析原理(英文版)M.北京：机械工业出版社.幂级数的展开及其应用在上一节中，我们讨论了幂级数的收敛性，在其收敛域内，幂级数总是收敛于一个和函数对于一些简单的幂级数，还可以借助逐项求导或求积分的方法，求出这个和函数本节将要讨论另外一个问题，对于任意一个函数，能否将其展开成一个幂级数，以及展开成的幂级数是否以为和函数?下面的讨论将解决这一问题一、 马克劳林(Macla
