1、Power Series Expansion and Its Applications 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.
2、This section will discuss another issue, for an arbitrary function()fx, can be expanded in a power series, and launched into. Whether the power series ()fx as and function? The following discussion will address this issue. 1 Maclaurin (Maclaurin) formula Polynomial power series can be seen as an ext
3、ension of reality, so consider the function ()fx can expand into power series, you can from the function ()fxand polynomials start to solve this problem. To this end, to give here without proof the following formula. Taylor (Taylor) formula, if the function ()fx at 0 xx in a neighborhood that until
4、the derivative of order 1n , then in the neighborhood of the following formula: 2 0000 ()()()()()() n n fxfxxxxxxxrx (9-5-1) Among 1 0 ()() n n rxxx That ( ) n rxfor the Lagrangian remainder. That (9-5-1)-type formula for the Taylor. If so 0 0x, get 2 ()( 0 )() n n fxfxxxrx, (9-5-2) At this point, (
5、1)(1) 11 1 ()() () (1)!(1)! nn nn n ffx rxxx nn (01). That (9-5-2) type formula for the Maclaurin. Formula shows that any function ()fxas long as until the 1n derivative, n can be equal to a polynomial and a remainder. We call the following power series () 2 (0)(0) ()(0)(0) 2 ! n n ff fxffxxx n (9-5
6、-3) For the Maclaurin series. So, is it to ()fx for the Sum functions? If the order Maclaurin series (9-5-3) the first 1n items and for 1( )n Sx , which () 2 1 (0)(0) ()(0)(0) 2 ! n n n ff Sxffxxx n Then, the series (9-5-3) converges to the function ()fx the conditions 1 lim()() n n sxfx . Noting Maclaurin formula (9-5-2) and the Maclaurin series (9-5-3) the relationship between the known 1 ( )( )( )
