In 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) formula
Polynomial power series can be seen as an extension of reality, 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.
在上一節(jié)中，我們討論了冪級(jí)數(shù)的收斂性，在其收斂域內(nèi)，冪級(jí)數(shù)總是收斂于一個(gè)和函數(shù)．對于一些簡單的冪級(jí)數(shù)，還可以借助逐項(xiàng)求導(dǎo)或求積分的方法，求出這個(gè)和函數(shù)．本節(jié)將要討論另外一個(gè)問題，對于任意一個(gè)函數(shù) ，能否將其展開成一個(gè)冪級(jí)數(shù)，以及展開成的冪級(jí)數(shù)是否以 為和函數(shù)?下面的討論將解決這一問題．
一、 馬克勞林(Maclaurin)公式
冪級(jí)數(shù)實(shí)際上可以視為多項(xiàng)式的延伸，因此在考慮函數(shù) 能否展開成冪級(jí)數(shù)時(shí)，可以從函數(shù) 與多項(xiàng)式的關(guān)系入手來解決這個(gè)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(shù) 在 的某一鄰域內(nèi)，有直到 階的導(dǎo)數(shù)，則在這個(gè)鄰域內(nèi)有如下公式：
,(951)