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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 this
This 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 example
The 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 estimate the value of .
Solution because
Because of
, ( ),
So there
Available 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.
References
1. A. K. Grunwald, Uber begrenzte Derivationen und deren Anwendung, Z. Angew. Math. Phys., 12 (1867), 441–480.
2. O. Heaviside, Electromagnetic Theory, vol. 2, Dover, 1950, Chap. 7, 8.
3. J. L. Lavoie, T. J. Osler and R. Trembley, Fractional derivatives and special functions, SIAM Rev., 18 (1976), 240–268.
4. A. V. Letnikov, Theory of differentiation of fractional order, Mat. Sb., 3 (1868), 1–68.
5. J. Liouville, Memoire sur quelques questions de géometrie et de mécanique, et surun noveau gentre pour resoudre ces questions, J. École Polytech., 13(1832), 1–69.
6. J. Liouville, Memoire: sur le calcul des differentielles á indices quelconques, J.École Polytech., 13(1832), 71–162.
7. A. C. McBride and G. F. Roach, Fractional Calculus, Pitman, 1985.
8. K. S. Miller Derivatives of noninteger order, Math. Mag., 68 (1995), 183–192.
9. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993.
10. P. A. Nekrassov, General differentiation, Mat. Sb., 14 (1888), 45–168.
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、 麦克劳林(Maclaurin)公式
幂级数实际上可以视为多项式的延伸，因此在考虑函数 能否展开成幂级数时，本文来自751/文(论"文?网，毕业论文 www.751com.cn 加7位QQ324~9114找原文 可以从函数 与多项式的关系入手来解决这个问题．为此，这里不加证明地给出如下的公式．
泰勒(Taylor)公式 如果函数 在 的某一邻域内，有直到 阶的导数，则在这个邻域内有如下公式：