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數(shù)學(xué)系英文論文翻譯(編輯修改稿)

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【文章內(nèi)容簡(jiǎn)介】 ries, 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 extension of reality, so consider the function ()fx can expand into power series, you can from the function ()fx and 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 0xx? in a neighborhood that until the derivative of order 1n? , then in the neighborhood of the following formula: 20 0 0 0( ) ( ) ( ) ( ) ( ) ( )n nf x f x x x x x x x r x? ? ? ? ? ? ? ? ?… (951) Among 10( ) ( )nnr x x x ??? That ()nrxfor the Lagrangian remainder. That (951)type formula for the Taylor. If so 0 0x? , get 2( ) ( 0) ( )n nf x f x x x r x? ? ? ? ? ?…, (952) At this point, ( 1 ) ( 1 )111 ( ) ( )() ( 1 ) ! ( 1 ) !nnnnn f f xr x x x??????? ?? (01???). That (952) type formula for the Maclaurin. Formula shows that any function ()fx as 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 nfff x f f x x xn???? ? ? ? ? ?… … (953) For the Maclaurin series. So, is it to ()fx for the Sum functions? If the order Maclaurin series (953) the first 1n? items and for 1()nSx? , which ()21 ( 0 ) ( 0 )( ) ( 0 ) ( 0 ) 2 ! !n nn ffS x f f x x xn? ???? ? ? ? ?… Then, the series (953) converges to the function ()fx the conditions 1lim ( ) ( )nn s x f x??? ?. Noting Maclaurin formula (952) and the Maclaurin series (953) the relationship between the known 1( ) ( ) ( )nnf x S x r x??? Thus, when ( ) 0nrx? There, 1( ) ( )nf x S x?? Vice versa. That if 1lim ( ) ( )nn s x f x??? ?, Units must ( ) 0nrx? . This indicates that the Maclaurin series (953) to ()fx and function as the Maclaurin formula

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