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 function and polynomials start to solve this problem. To this end, to give here without proof the following formula.Taylor (Taylor) formula, if the functionatin a neighborhood that until the derivative of order , then in t

4、he neighborhood of the following formula:(9-5-1)AmongThat 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 polynom

5、ial 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 Mac

6、laurin 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

7、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

8、 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

9、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 meth

10、od.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

11、 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, alt

12、hough 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 achie

13、ve many functions of power series expansion. This demand function of power series expansion method is called indirect expansion.Example 2Find the function,Department in the power series expansion.Solutionbecause,And，（）Therefore, the power series can be itemized according to the rules of derivation c

14、an 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

15、 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.此外文文獻選自于：Walter.Rudin.數學分析原理(英文版)M.北京：機械工業出版社.冪級數的展開及其應用在上一節中，我們討論了冪級數的收斂性，在其收斂域內，

16、冪級數總是收斂于一個和函數對于一些簡單的冪級數，還可以借助逐項求導或求積分的方法，求出這個和函數本節將要討論另外一個問題，對于任意一個函數，能否將其展開成一個冪級數，以及展開成的冪級數是否以為和函數?下面的討論將解決這一問題一、 馬克勞林(Maclaurin)公式冪級數實際上可以視為多項式的延伸，因此在考慮函數能否展開成冪級數時，可以從函數與多項式的關系入手來解決這個問題為此，這里不加證明地給出如下的公式泰勒(Taylor)公式 如果函數在的某一鄰域內，有直到階的導數，則在這個鄰域內有如下公式：,(9-5-1)其中稱為拉格朗日型余項稱(9-5-1)式為泰勒公式如果令，就得到， (9-5-2)此

17、時，, ()稱(9-5-2)式為馬克勞林公式公式說明，任一函數只要有直到階導數，就可等于某個次多項式與一個余項的和我們稱下列冪級數 (9-5-3)為馬克勞林級數那么，它是否以為和函數呢?若令馬克勞林級數(9-5-3)的前項和為，即，那么，級數(9-5-3)收斂于函數的條件為注意到馬克勞林公式(9-5-2)與馬克勞林級數(9-5-3)的關系，可知于是，當時，有反之亦然即若則必有這表明，馬克勞林級數(9-5-3)以為和函數馬克勞林公式(9-5-2)中的余項 (當時)這樣，我們就得到了函數的冪級數展開式：(9-5-4)它就是函數的冪級數表達式，也就是說，函數的冪級數展開式是唯一的事實上，假設函數可以

18、表示為冪級數， (9-5-5)那么，根據冪級數在收斂域內可逐項求導的性質，再令(冪級數顯然在點收斂)，就容易得到將它們代入(9-5-5)式，所得與的馬克勞林展開式(9-5-4)完全相同綜上所述，如果函數在包含零的某區間內有任意階導數，且在此區間內的馬克勞林公式中的余項以零為極限(當時)，那么，函數就可展開成形如(9-5-4)式的冪級數冪級數,稱為泰勒級數二、 初等函數的冪級數展開式利用馬克勞林公式將函數展開成冪級數的方法，稱為直接展開法例1 試將函數展開成的冪級數解 因為,所以，于是我們得到冪級數， (9-5-6)顯然，(9-5-6)式的收斂區間為，至于(9-5-6)式是否以為和函數，即它是否收斂于，還要考察余項因為 ()，且，所以注意到對任一確定的值，是一個確定的常數，而級數(9-5-6)是絕對收斂的，因此其一般項當時，所以當時，有，由此可知這表明級數(9-5-6)確實收斂于，因此有 ()這種運用馬克勞林公式將函數展開成冪級數的方法，雖然程序明確，但是運算往往過于繁瑣，因此人們普遍采用下面的比較簡便的冪級數展開法在此之前，我們已經得到了函